1.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
2.
Area
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Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the surface of a three-dimensional object. It is the analog of the length of a curve or the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size, in the International System of Units, the standard unit of area is the square metre, which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the area as three such squares. In mathematics, the square is defined to have area one. There are several formulas for the areas of simple shapes such as triangles, rectangles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles, for shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a motivation for the historical development of calculus. For a solid such as a sphere, cone, or cylinder. Formulas for the areas of simple shapes were computed by the ancient Greeks. Area plays an important role in modern mathematics, in addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, in general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers and it can be proved that such a function exists. An approach to defining what is meant by area is through axioms, area can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties, For all S in M, a ≥0. If S and T are in M then so are S ∪ T and S ∩ T, if S and T are in M with S ⊆ T then T − S is in M and a = a − a. If a set S is in M and S is congruent to T then T is also in M, every rectangle R is in M. If the rectangle has length h and breadth k then a = hk, let Q be a set enclosed between two step regions S and T
3.
Surface (mathematics)
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In mathematics, a surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line, there are several more precise definitions, depending on the context and the mathematical tools that are used for the study. Often, a surface is defined by equations that are satisfied by the coordinates of its points and this is the case of the graph of a continuous function of two variables. The set of the zeros of a function of three variables is a surface, which is called an implicit surface, if the defining three-variate function is a polynomial, the surface is an algebraic surface. For example, the sphere is an algebraic surface, as it may be defined by the implicit equation x 2 + y 2 + z 2 −1 =0. A surface may also be defined as the image, in space of dimension at least 3. In this case, one says that one has a parametric surface, for example, the unit sphere may be parametrized by the Euler angles, also called longitude u and latitude v by x = cos cos y = sin cos z = sin . Parametric equations of surfaces are often irregular at some points, for example, all but two points of the unit sphere, are the image, by the above parametrization, of exactly one pair of Euler angles. For the remaining two points, one has cos v =0, and the longitude u may take any values, also, there are surfaces for which there cannot exits a single parametrization that covers the whole surface. Therefore, one often considers surfaces which are parametrized by several parametric equations and this allows defining surfaces in spaces of dimension higher than three, and even abstract surfaces, which are not contained in any other space. On the other hand, this excludes surfaces that have singularities, in classical geometry, a surface is generally defined as a locus of a point or a line. A ruled surface is the locus of a moving line satisfying some constraints, in modern terminology, a surface is a surface. In this article, several kinds of surfaces are considered and compared, a non-ambiguous terminology is thus necessary for distinguish them. Therefore, we call topological surfaces the surfaces that are manifolds of dimension two and we call differential surfaces the surfaces that are differentiable manifolds. Every differential surface is a surface, but the converse is false. For simplicity, unless stated, surface will mean a surface in the Euclidean space of dimension 3 or in R3. A surface, that is not supposed to be included in another space, is called an abstract surface, the graph of a continuous function of two variables, defined over a connected open subset of R2 is a topological surface. If the function is differentiable, the graph is a differential surface, a plane is together an algebraic surface and a differentiable surface
4.
Arc length
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Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves, the advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases. A curve in the plane can be approximated by connecting a number of points on the curve using line segments to create a polygonal path. If the curve is not already a polygonal path, using a larger number of segments of smaller lengths will result in better approximations. For some curves there is a smallest number L that is a bound on the length of any polygonal approximation. These curves are called rectifiable and the number L is defined as the arc length, let f, → R n be a continuously differentiable function. The length of the curve defined by f can be defined as the limit of the sum of line segment lengths for a partition of as the number of segments approaches infinity. This means L = lim N → ∞ ∑ i =1 N | f − f | where t i = a + i / N = a + i Δ t for i =0,1, …, N. This means ∑ i =1 N | f − f Δ t | Δ t − ∑ i =1 N | f ′ | Δ t has absolute value less than ϵ for N > / δ. This means that in the limit N → ∞, the left term above equals the right term and this definition of arc length shows that the length of a curve f, → R n continuously differentiable on is always finite. In other words, the curve is always rectifiable and this definition is also valid if f is merely continuous, not differentiable. A curve can be parameterized in infinitely many ways, let φ, → be any continuously differentiable bijection. Then g = f ∘ φ −1, → R n is another continuously differentiable parameterization of the curve defined by f. Curves with closed-form solutions for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola, the lack of a closed form solution for the arc length of an elliptic arc led to the development of the elliptic integrals. In most cases, including even simple curves, there are no solutions for arc length. Numerical integration of the arc length integral is very efficient. For example, consider the problem of finding the length of a quarter of the circle by numerically integrating the arc length integral. The upper half of the circle can be parameterized as y =1 − x 2
5.
Polyhedron
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In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra
6.
Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
7.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
8.
Partial derivative
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In mathematics, the symmetry of second derivatives refers to the possibility under certain conditions of interchanging the order of taking partial derivatives of a function f of n variables. This is sometimes known as Schwarzs theorem or Youngs theorem, in the context of partial differential equations it is called the Schwarz integrability condition. This matrix of partial derivatives of f is called the Hessian matrix of f. The entries in it off the diagonal are the mixed derivatives. In most real-life circumstances the Hessian matrix is symmetric, although there are a number of functions that do not have this property. Mathematical analysis reveals that symmetry requires a hypothesis on f that goes further than simply stating the existence of the derivatives at a particular point. Schwarz theorem gives a sufficient condition on f for this to occur, in symbols, the symmetry says that, for example, ∂ ∂ x = ∂ ∂ y. This equality can also be written as ∂ x y f = ∂ y x f, alternatively, the symmetry can be written as an algebraic statement involving the differential operator Di which takes the partial derivative with respect to xi, Di. From this relation it follows that the ring of operators with constant coefficients. But one should naturally specify some domain for these operators and it is easy to check the symmetry as applied to monomials, so that one can take polynomials in the xi as a domain. In fact smooth functions are possible, the partial differentiations of this function are commutative at that point. One easy way to establish this theorem is by applying Greens theorem to the gradient of f, a weaker condition than the continuity of second partial derivatives which nevertheless suffices to ensure symmetry is that all partial derivatives are themselves differentiable. The theory of distributions eliminates analytic problems with the symmetry, the derivative of an integrable function can always be defined as a distribution, and symmetry of mixed partial derivatives always holds as an equality of distributions. The use of integration by parts to define differentiation of distributions puts the symmetry question back onto the test functions. In more detail, = − = f = f = − =, another approach, which defines the Fourier transform of a function, is to note that on such transforms partial derivatives become multiplication operators that commute much more obviously. The symmetry may be if the function fails to have differentiable partial derivatives. An example of non-symmetry is the function, This function is everywhere continuous, however, the second partial derivatives are not continuous at, and the symmetry fails. In fact, along the x-axis the y-derivative is ∂ y f | = x, vice versa, along the y-axis the x-derivative ∂ x f | = − y, and so ∂ y ∂ x f | = −1
9.
Multiple integral
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The multiple integral is a collective term for the definite integral of functions of more than one real variable, for example, f or f. Integrals of a function of two variables over a region in R2 are called double integrals, and integrals of a function of three variables over a region of R3 are called triple integrals, if there are more variables, a multiple integral will yield hypervolumes of multidimensional functions. For n >1, consider a so-called half-open n-dimensional hyperrectangular domain T, defined as, partition each interval [aj, bj) into a finite family Ij of non-overlapping subintervals ijα, with each subinterval closed at the left end, and open at the right end. Then the finite family of subrectangles C given by C = I1 × I2 × ⋯ × I n is a partition of T, that is, let f, T → R be a function defined on T. The diameter of a subrectangle Ck is the largest of the lengths of the intervals whose Cartesian product is Ck, the diameter of a given partition of T is defined as the largest of the diameters of the subrectangles in the partition. Intuitively, as the diameter of the partition C is restricted smaller and smaller, the number of subrectangles m gets larger, and the measure m of each subrectangle grows smaller. Then the integral of the function over the original domain is defined to be the integral of the extended function over its rectangular domain. In what follows the Riemann integral in n dimensions will be called the multiple integral, multiple integrals have many properties common to those of integrals of functions of one variable. One important property of multiple integrals is that the value of an integral is independent of the order of integrands under certain conditions and this property is popularly known as Fubinis theorem. For continuous functions, this is justified by Fubinis theorem, sometimes, it is possible to obtain the result of the integration by direct examination without any calculations. The following are some methods of integration, When the integrand is a constant function c, the integral is equal to the product of c. If c =1 and the domain is a subregion of R2, the integral gives the area of the region, while if the domain is a subregion of R3, the integral gives the volume of the region. When the integrand is even with respect to this variable, the integral is equal to twice the integral over one half of the domain, as the integrals over the two halves of the domain are equal. Consider the function f =2 sin − 3y3 +5 integrated over the domain T =, a disc with radius 1 centered at the origin with the boundary included. Similarly, the function 3y3 is an odd function of y, and T is symmetric with respect to the x-axis, therefore the original integral is equal to the area of the disk times 5, or 5π. Consider the function f = x exp and as integration region the sphere with radius 2 centered at the origin, T =. The ball is symmetric about all three axes, but it is sufficient to integrate with respect to x-axis to show that the integral is 0, such a domain will be here called a normal domain. Elsewhere in the literature, normal domains are called type I or type II domains
10.
Henri Lebesgue
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His theory was published originally in his dissertation Intégrale, longueur, aire at the University of Nancy during 1902. Henri Lebesgue was born on 28 June 1875 in Beauvais, Oise, Lebesgues father was a typesetter and his mother was a school teacher. His parents assembled at home a library that the young Henri was able to use and his father died of tuberculosis when Lebesgue was still very young and his mother had to support him by herself. In 1894 Lebesgue was accepted at the École Normale Supérieure, where he continued to focus his energy on the study of mathematics, graduating in 1897. At the same time he started his studies at the Sorbonne. In 1899 he moved to a position at the Lycée Central in Nancy. In 1902 he earned his Ph. D. from the Sorbonne with the thesis on Integral, Length, Area, submitted with Borel, four years older. Lebesgue married the sister of one of his students, and he. After publishing his thesis, Lebesgue was offered in 1902 a position at the University of Rennes, lecturing there until 1906, in 1910 Lebesgue moved to the Sorbonne as a maître de conférences, being promoted to professor starting with 1919. In 1921 he left the Sorbonne to become professor of mathematics at the Collège de France, in 1922 he was elected a member of the Académie des Sciences. Henri Lebesgue died on 26 July 1941 in Paris, Lebesgues first paper was published in 1898 and was titled Sur lapproximation des fonctions. It dealt with Weierstrass theorem on approximation to continuous functions by polynomials, between March 1899 and April 1901 Lebesgue published six notes in Comptes Rendus. The first of these, unrelated to his development of Lebesgue integration, Lebesgues great thesis, Intégrale, longueur, aire, with the full account of this work, appeared in the Annali di Matematica in 1902. The first chapter develops the theory of measure, in the second chapter he defines the integral both geometrically and analytically. The next chapters expand the Comptes Rendus notes dealing with length, area, the final chapter deals mainly with Plateaus problem. This dissertation is considered to be one of the finest ever written by a mathematician and his lectures from 1902 to 1903 were collected into a Borel tract Leçons sur lintégration et la recherche des fonctions primitives. The problem of integration regarded as the search for a function is the keynote of the book. Lebesgue presents the problem of integration in its context, addressing Augustin-Louis Cauchy, Peter Gustav Lejeune Dirichlet
11.
Hermann Minkowski
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Hermann Minkowski was a Jewish German mathematician, professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used methods to solve problems in number theory, mathematical physics. Hermann was a brother of the medical researcher, Oskar. In different sources Minkowskis nationality is given as German, Polish, Lithuanian or Lithuanian-German. To escape persecution in Russia the family moved to Königsberg in 1872, Minkowski studied in Königsberg and taught in Bonn, Königsberg and Zurich, and finally in Göttingen from 1902 until his premature death in 1909. He married Auguste Adler in 1897 with whom he had two daughters, the engineer and inventor Reinhold Rudenberg was his son-in-law. Minkowski died suddenly of appendicitis in Göttingen on 12 January 1909 and our science, which we loved above all else, brought us together, it seemed to us a garden full of flowers. He was for me a gift from heaven and I must be grateful to have possessed that gift for so long. Now death has suddenly torn him from our midst, however, what death cannot take away is his noble image in our hearts and the knowledge that his spirit continues to be active in us. The main-belt asteroid 12493 Minkowski and M-matrices are named in Minkowskis honor, Minkowski was educated in Germany at the Albertina University of Königsberg, where he earned his doctorate in 1885 under the direction of Ferdinand von Lindemann. In 1883, while still a student at Königsberg, he was awarded the Mathematics Prize of the French Academy of Sciences for his manuscript on the theory of quadratic forms and he also became a friend of another renowned mathematician, David Hilbert. His brother, Oskar Minkowski, was a physician and researcher. Minkowski taught at the universities of Bonn, Göttingen, Königsberg, at the Eidgenössische Polytechnikum, today the ETH Zurich, he was one of Einsteins teachers. Minkowski explored the arithmetic of quadratic forms, especially concerning n variables, in 1896, he presented his geometry of numbers, a geometrical method that solved problems in number theory. He is also the creator of the Minkowski Sausage and the Minkowski cover of a curve, in 1902, he joined the Mathematics Department of Göttingen and became a close colleague of David Hilbert, whom he first met at university in Königsberg. Constantin Carathéodory was one of his students there, henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 53–111. English translation, The Fundamental Equations for Electromagnetic Processes in Moving Bodies, in, The Principle of Relativity, Calcutta, University Press, 1–69 Minkowski, Hermann
12.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
13.
Surface (topology)
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In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an abstract surface not embedded in any Euclidean space. For example, the Klein bottle is a surface, which cannot be represented in the three-dimensional Euclidean space without introducing self-intersections, in mathematics, a surface is a geometrical shape that resembles to a deformed plane. The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R3, the exact definition of a surface may depend on the context. Typically, in geometry, a surface may cross itself, while, in topology and differential geometry. A surface is a space, this means that a moving point on a surface may move in two directions. In other words, around almost every point, there is a patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles a two-dimensional sphere, the concept of surface is widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the properties of an airplane. A surface is a space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E2. Such a neighborhood, together with the corresponding homeomorphism, is known as a chart and it is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane. These coordinates are known as coordinates and these homeomorphisms lead us to describe surfaces as being locally Euclidean. In most writings on the subject, it is assumed, explicitly or implicitly, that as a topological space a surface is also nonempty, second countable. It is also assumed that the surfaces under consideration are connected. The rest of this article will assume, unless specified otherwise, that a surface is nonempty, Hausdorff, second countable and these homeomorphisms are also known as charts. The boundary of the upper half-plane is the x-axis, a point on the surface mapped via a chart to the x-axis is termed a boundary point. The collection of points is known as the boundary of the surface which is necessarily a one-manifold, that is. On the other hand, a point mapped to above the x-axis is an interior point, the collection of interior points is the interior of the surface which is always non-empty. The closed disk is an example of a surface with boundary
14.
Congruence (geometry)
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In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. This means that either object can be repositioned and reflected so as to coincide precisely with the other object, so two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted, in elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects, two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure, two circles are congruent if they have the same diameter. The related concept of similarity applies if the objects differ in size, for two polygons to be congruent, they must have an equal number of sides. Two polygons with n sides are congruent if and only if they each have identical sequences side-angle-side-angle-. for n sides. Congruence of polygons can be established graphically as follows, First, match, second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches, fourth, reflect the rotated figure about this matched side until the figures match. If at any time the step cannot be completed, the polygons are not congruent, two triangles are congruent if their corresponding sides are equal in length, in which case their corresponding angles are equal in measure. SSS, If three pairs of sides of two triangles are equal in length, then the triangles are congruent, ASA, If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA Postulate was contributed by Thales of Miletus, in most systems of axioms, the three criteria—SAS, SSS and ASA—are established as theorems. In the School Mathematics Study Group system SAS is taken as one of 22 postulates, AAS, If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. For American usage, AAS is equivalent to an ASA condition, RHS, also known as HL, If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent. The SSA condition which specifies two sides and a non-included angle does not by itself prove congruence, in order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. The opposite side is longer when the corresponding angles are acute. This is the case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence
15.
Differentiable function
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In calculus, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a function must have a tangent line at each point in its domain, be relatively smooth. More generally, if x0 is a point in the domain of a function f and this means that the graph of f has a non-vertical tangent line at the point. The function f may also be called locally linear at x0, if f is differentiable at a point x0, then f must also be continuous at x0. In particular, any function must be continuous at every point in its domain. The converse does not hold, a function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, most functions that occur in practice have derivatives at all points or at almost every point. However, a result of Stefan Banach states that the set of functions that have a derivative at some point is a set in the space of all continuous functions. Informally, this means that differentiable functions are very atypical among continuous functions, the first known example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function. A function f is said to be continuously differentiable if the derivative f exists and is itself a continuous function, though the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity. For example, the function f = { x 2 sin if x ≠00 if x =0 is differentiable at 0, since f ′ = lim ϵ →0 =0, exists. However, for x≠0, f ′ =2 x sin − cos which has no limit as x →0, nevertheless, Darbouxs theorem implies that the derivative of any function satisfies the conclusion of the intermediate value theorem. Sometimes continuously differentiable functions are said to be of class C1, a function is of class C2 if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of class Ck if the first k derivatives f′, F all exist and are continuous. If derivatives f exist for all integers n, the function is smooth or equivalently. If all the derivatives of a function exist and are continuous in a neighborhood of a point, then the function is differentiable at that point. If a function is differentiable at x0, then all of the partial derivatives exist at x0, a similar formulation of the higher-dimensional derivative is provided by the fundamental increment lemma found in single-variable calculus. Note that existence of the partial derivatives does not in general guarantee that a function is differentiable at a point
16.
Surface of revolution
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A surface of revolution is a surface in Euclidean space created by rotating a curve around an axis of rotation. Examples of surfaces of revolution generated by a line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. The sections of the surface of revolution made by planes through the axis are called meridional sections, any meridional section can be considered to be the generatrix in the plane determined by it and the axis. The sections of the surface of revolution made by planes that are perpendicular to the axis are circles, some special cases of hyperboloids and elliptic paraboloids are surfaces of revolution. These may be identified as those quadratic surfaces all of whose cross sections perpendicular to the axis are circular and this formula is the calculus equivalent of Pappuss centroid theorem. The quantity 2 +2 comes from the Pythagorean theorem and represents a segment of the arc of the curve. The quantity 2πx is the path of this segment, as required by Pappus theorem. Likewise, when the axis of rotation is the x-axis and provided that y is never negative and these come from the above formula. For example, the surface with unit radius is generated by the curve y = sin, x = cos. Its area is therefore A =2 π ∫0 π sin 2 +2 d t =2 π ∫0 π sin d t =4 π. A basic problem in the calculus of variations is finding the curve between two points that produces this surface of revolution. There are only two minimal surfaces of revolution, the plane and the catenoid, geodesics on a surface of revolution are governed by Clairauts relation. A surface of revolution with a hole in, where the axis of revolution does not intersect the surface, is called a toroid, for example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow square-section ring is produced. If the revolved figure is a circle, then the object is called a torus, the use of surfaces of revolution is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to surface area without the use of measuring the length
17.
Hermann Schwarz
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Karl Hermann Amandus Schwarz was a German mathematician, known for his work in complex analysis. Schwarz was born in Hermsdorf, Silesia and he was married to Marie Kummer, who was the daughter to the mathematician Ernst Eduard Kummer and Ottilie née Mendelssohn. Schwarz and Kummer had six children, Schwarz originally studied chemistry in Berlin but Ernst Eduard Kummer and Karl Theodor Wihelm Weierstraß persuaded him to change to mathematics. He received his Ph. D. from the Universität Berlin in 1864 and was advised by Ernst Kummer, between 1867 and 1869 he worked at the University of Halle, then at the Swiss Federal Polytechnic. From 1875 he worked at Göttingen University, dealing with the subjects of analysis, differential geometry. Schwarzs works include Bestimmung einer speziellen Minimalfläche, which was crowned by the Berlin Academy in 1867 and printed in 1871 and his work on the latter allowed Émile Picard to show solutions of differential equations exist. In 1892 he became a member of the Berlin Academy of Science and a professor at the University of Berlin, in total, he advised 20 Ph. D students. His name is attached to many ideas in mathematics, including, Schwarz, H. A. Bestimmung einer speziellen Minimalfläche, Dümmler Schwarz, H. A. Gesammelte mathematische Abhandlungen. AMS Chelsea Publishing, ISBN 978-0-8284-0260-6, MR0392470 OConnor, John J. Robertson, Edmund F. Hermann Schwarz, MacTutor History of Mathematics archive, Hermann Schwarz at the Mathematics Genealogy Project
18.
Fractal
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A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known as expanding symmetry or evolving symmetry, if the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge, Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set, Fractals also include the idea of a detailed pattern that repeats itself. Fractals are different from other geometric figures because of the way in which they scale, doubling the edge lengths of a polygon multiplies its area by four, which is two raised to the power of two. Likewise, if the radius of a sphere is doubled, its volume scales by eight, but if a fractals one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the dimension of the fractal. As mathematical equations, fractals are usually nowhere differentiable, the term fractal was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning broken or fractured, there is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as beautiful, damn hard, increasingly useful, Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractal. The word fractal often has different connotations for laypeople than for mathematicians, the mathematical concept is difficult to define formally even for mathematicians, but key features can be understood with little mathematical background. If this is done on fractals, however, no new detail appears, nothing changes, self-similarity itself is not necessarily counter-intuitive. The difference for fractals is that the pattern reproduced must be detailed, a regular line, for instance, is conventionally understood to be 1-dimensional, if such a curve is divided into pieces each 1/3 the length of the original, there are always 3 equal pieces. In contrast, consider the Koch snowflake and it is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured. This also leads to understanding a third feature, that fractals as mathematical equations are nowhere differentiable, in a concrete sense, this means fractals cannot be measured in traditional ways. The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, according to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity. In his writings, Leibniz used the term fractional exponents, also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called self-inverse fractals
19.
Cube
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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids and it has 6 faces,12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and it is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron and it has cubical or octahedral symmetry. The cube has four special orthogonal projections, centered, on a vertex, edges, face, the first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of length a, As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares. A cube has the largest volume among cuboids with a surface area. Also, a cube has the largest volume among cuboids with the same linear size. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the faces around each vertex,111,112,123. The cube has three classes of symmetry, which can be represented by coloring the faces. The highest octahedral symmetry Oh has all the faces the same color, the dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a symmetry, with sides alternating colors. Each symmetry form has a different Wythoff symbol, a cube has eleven nets, that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the color, one would need at least three colors
20.
Cuboid
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In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. By Eulers formula the numbers of faces F, of vertices V, in the case of a cuboid this gives 6 +8 =12 +2, that is, like a cube, a cuboid has 6 faces,8 vertices, and 12 edges. Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, in a rectangular cuboid, all angles are right angles, and opposite faces of a cuboid are equal. By definition this makes it a rectangular prism, and the terms rectangular parallelepiped or orthogonal parallelepiped are also used to designate this polyhedron. The terms rectangular prism and oblong prism, however, are ambiguous, the square cuboid, square box, or right square prism is a special case of the cuboid in which at least two faces are squares. It has Schläfli symbol ×, and its symmetry is doubled from to, the cube is a special case of the square cuboid in which all six faces are squares. It has Schläfli symbol, and its symmetry is raised from, to, if the dimensions of a rectangular cuboid are a, b and c, then its volume is abc and its surface area is 2. The length of the diagonal is d = a 2 + b 2 + c 2. Cuboid shapes are used for boxes, cupboards, rooms, buildings. Cuboids are among those solids that can tessellate 3-dimensional space, the shape is fairly versatile in being able to contain multiple smaller cuboids, e. g. sugar cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building. A cuboid with integer edges as well as integer face diagonals is called an Euler brick, a perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists, the number of different nets for a simple cube is 11, however this number increases significantly to 54 for a rectangular cuboid of 3 different lengths. Hyperrectangle Trapezohedron Weisstein, Eric W. Cuboid, rectangular prism and cuboid Paper models and pictures
21.
Triangular prism
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In geometry, a triangular prism is a three-sided prism, it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is oblique, a uniform triangular prism is a right triangular prism with equilateral bases, and square sides. Equivalently, it is a polyhedron of which two faces are parallel, while the normals of the other three are in the same plane. All cross-sections parallel to the faces are the same triangle. A right triangular prism is semiregular or, more generally, a uniform if the base faces are equilateral triangles. It can be seen as a truncated trigonal hosohedron, represented by Schläfli symbol t, alternately it can be seen as the Cartesian product of a triangle and a line segment, and represented by the product x. The dual of a prism is a triangular bipyramid. The symmetry group of a right 3-sided prism with triangular base is D3h of order 12, the rotation group is D3 of order 6. The symmetry group does not contain inversion, the volume of any prism is the product of the area of the base and the distance between the two bases. A truncated right triangular prism has one triangular face truncated at an oblique angle, there are two full D2h symmetry facetings of a triangular prism, both with 6 isosceles triangle faces, one keeping the original top and bottom triangles, and one the original squares. Two lower C3v symmetry faceting have one triangle,3 lateral crossed square faces. This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations and this polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure, and continues as tilings of the hyperbolic plane. These vertex-transitive figures have reflectional symmetry and this polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure, and continues as tilings of the hyperbolic plane. These vertex-transitive figures have reflectional symmetry, there are 4 uniform compounds of triangular prisms, Compound of four triangular prisms, compound of eight triangular prisms, compound of ten triangular prisms, compound of twenty triangular prisms. Each progressive uniform polytope is constructed vertex figure of the previous polytope, thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes. In Coxeters notation the triangular prism is given the symbol −121, the triangular prism exists as cells of a number of four-dimensional uniform 4-polytopes, including, Wedge Weisstein, Eric W. Triangular prism. Interactive Polyhedron, Triangular Prism surface area and volume of a triangular prism
22.
Prism (geometry)
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In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases, prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids, a right prism is a prism in which the joining edges and faces are perpendicular to the base faces. This applies if the faces are rectangular. If the joining edges and faces are not perpendicular to the base faces, for example a parallelepiped is an oblique prism of which the base is a parallelogram, or equivalently a polyhedron with six faces which are all parallelograms. A truncated prism is a prism with nonparallel top and bottom faces, some texts may apply the term rectangular prism or square prism to both a right rectangular-sided prism and a right square-sided prism. A right p-gonal prism with rectangular sides has a Schläfli symbol ×, a right rectangular prism is also called a cuboid, or informally a rectangular box. A right square prism is simply a box, and may also be called a square cuboid. A right rectangular prism has Schläfli symbol ××, an n-prism, having regular polygon ends and rectangular sides, approaches a cylindrical solid as n approaches infinity. The term uniform prism or semiregular prism can be used for a prism with square sides. A uniform p-gonal prism has a Schläfli symbol t, right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra, the other series being the antiprisms. The dual of a prism is a bipyramid. The volume of a prism is the product of the area of the base, the volume is therefore, V = B ⋅ h where B is the base area and h is the height. The volume of a prism whose base is a regular n-sided polygon with side s is therefore. The surface area of a prism is 2 · B + P · h, where B is the area of the base, h the height. The surface area of a prism whose base is a regular n-sided polygon with side length s and height h is therefore. The rotation group is Dn of order 2n, except in the case of a cube, which has the symmetry group O of order 24. The symmetry group Dnh contains inversion iff n is even, a prismatic polytope is a higher-dimensional generalization of a prism
23.
Spherical lune
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In spherical geometry, a spherical lune is an area on a sphere bounded by two half great circles which meet at antipodal points, and is an example of a digon, θ, with dihedral angle θ. The word lune derives from luna, the Latin word for Moon, a spherical wedge is the volume of space bounded by two planes passing through a sphere center and the surface of the sphere. Great circles are the largest possible circles of a sphere, each one divides the surface of the sphere into two equal halves, two great circles always intersect at two polar opposite points. Common examples of great circles are lines of longitude on a sphere, a spherical lune has two planes of symmetry. It can be bisected into two lunes of half the angle, or it can be bisected by a line into two right spherical triangles. The surface area of a spherical lune is 2θ R2, where R is the radius of the sphere, when this angle equals 2π radians — i. e. A hosohedron is a tessellation of the sphere by lunes, a n-gonal regular hosohedron, has n equal lunes of π/n radians. An n-hosohedron has dihedral symmetry Dnh, of order 4n, each lune individually has cyclic symmetry C2v, of order 4. Each hosohedra can be divided by an equatorial bisector into two spherical triangles. The visibly lighted portion of the Moon visible from the Earth is a spherical lune, the first of the two intersecting great circles is the terminator between the sunlit half of the Moon and the dark half. The second great circle is a terrestrial terminator that separates the half visible from the Earth from the unseen half, the spherical lune is a lighted crescent shape seen from Earth. Lunes can be defined on higher dimensional spheres as well, in 4-dimensions a 3-sphere is a generalized sphere. It can contain regular digon lunes as θ, φ, where θ and φ are two dihedral angles, for example, a regular hosotope has digon faces, 2π/p, 2π/q, where its vertex figure is a spherical platonic solid. Each vertex of defines an edge in the hosotope and adjacent pairs of those edges define lune faces, beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL, CRC Press, p.130,1987. Harris, J. W. and Stocker, H, §4.8.6 in Handbook of Mathematics and Computational Science. New York, Springer-Verlag, p.108,1998, gellert, W. Gottwald, S. Hellwich, M. Kästner, H. and Künstner, H. VNR Concise Encyclopedia of Mathematics, 2nd ed. New York, Van Nostrand Reinhold, p.262,1989
24.
Dihedral angle
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A dihedral angle is the angle between two intersecting planes. In chemistry it is the angle between planes through two sets of three atoms, having two atoms in common, in solid geometry it is defined as the union of a line and two half-planes that have this line as a common edge. In higher dimension, a dihedral angle represents the angle between two hyperplanes, a dihedral angle is an angle between two intersecting planes on a third plane perpendicular to the line of intersection. A torsion angle is an example of a dihedral angle. In stereochemistry every set of three atoms of a molecule defines a plane, when two such planes intersect, the angle between them is a dihedral angle. Dihedral angles are used to specify the molecular conformation, stereochemical arrangements corresponding to angles between 0° and ±90° are called syn, those corresponding to angles between ±90° and 180° anti. Similarly, arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called clinal and those between 0° and ±30° or ±150° and 180° are called periplanar. The synperiplanar conformation is also known as the syn- or cis-conformation, antiperiplanar as anti or trans, for example, with n-butane two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms. The syn-conformation shown above, with an angle of 60° is less stable than the anti-configuration with a dihedral angle of 180°. For macromolecular usage the symbols T, C, G+, G−, A+, a Ramachandran plot, originally developed in 1963 by G. N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan, is a way to visualize energetically allowed regions for backbone dihedral angles ψ against φ of amino acid residues in protein structure, the figure at right illustrates the definition of the φ and ψ backbone dihedral angles. In a protein chain three dihedral angles are defined as φ, ψ and ω, as shown in the diagram, the planarity of the peptide bond usually restricts ω to be 180° or 0°. The distance between the Cα atoms in the trans and cis isomers is approximately 3.8 and 2.9 Å, the cis isomer is mainly observed in Xaa–Pro peptide bonds. The sidechain dihedral angles tend to cluster near 180°, 60°, and −60°, which are called the trans, gauche+, the stability of certain sidechain dihedral angles is affected by the values φ and ψ. For instance, there are steric interactions between the Cγ of the side chain in the gauche+ rotamer and the backbone nitrogen of the next residue when ψ is near -60°. An alternative method is to calculate the angle between the vectors, nA and nB, which are normal to the planes. Cos φ = − n A ⋅ n B | n A | | n B | where nA · nB is the dot product of the vectors and |nA| |nB| is the product of their lengths. Any plane can also be described by two non-collinear vectors lying in that plane, taking their cross product yields a vector to the plane
25.
Torus
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In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a shape and is called a torus of revolution. Real-world examples of objects include inner tubes, swim rings, and the surface of a doughnut. A torus should not be confused with a solid torus, which is formed by rotating a disc, rather than a circle, a solid torus is a torus plus the volume inside the torus. Real-world approximations include doughnuts, vadai or vada, many lifebuoys, in topology, a ring torus is homeomorphic to the Cartesian product of two circles, S1 × S1, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1, the ring torus is one way to embed this space into three-dimensional Euclidean space, but another way to do this is the Cartesian product of the embedding of S1 in the plane. This produces an object called the Clifford torus, a surface in 4-space. In the field of topology, a torus is any space that is topologically equivalent to a torus. R is known as the radius and r is known as the minor radius. The ratio R divided by r is known as the aspect ratio, a doughnut has an aspect ratio of about 2 to 3. An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is 2 + z 2 = r 2, or the solution of f =0, algebraically eliminating the square root gives a quartic equation,2 =4 R2. The three different classes of standard tori correspond to the three aspect ratios between R and r, When R > r, the surface will be the familiar ring torus. R = r corresponds to the torus, which in effect is a torus with no hole. R < r describes the self-intersecting spindle torus, when R =0, the torus degenerates to the sphere. When R ≥ r, the interior 2 + z 2 < r 2 of this torus is diffeomorphic to a product of an Euclidean open disc, the losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side. In traditional spherical coordinates there are three measures, R, the distance from the center of the system, and θ and φ. As a torus has, effectively, two points, the centerpoints of the angles are moved, φ measures the same angle as it does in the spherical system. The center point of θ is moved to the center of r and these terms were first used in a discussion of the Earths magnetic field, where poloidal was used to denote the direction toward the poles
26.
Cylinder
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In its simplest form, a cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes, commonly the word cylinder is understood to refer to a finite section of a right circular cylinder having a finite height with circular ends perpendicular to the axis as shown in the figure. If the ends are open, it is called an open cylinder, if the ends are closed by flat surfaces it is called a solid cylinder. The formulae for the area and the volume of such a cylinder have been known since deep antiquity. The area of the side is known as the lateral area. An open cylinder does not include either top or bottom elements, the surface area of a closed cylinder is made up the sum of all three components, top, bottom and side. Its surface area is A = 2πr2 + 2πrh = 2πr = πd=L+2B, for a given volume, the closed cylinder with the smallest surface area has h = 2r. Equivalently, for a surface area, the closed cylinder with the largest volume has h = 2r. Cylindric sections are the intersections of cylinders with planes, for a right circular cylinder, there are four possibilities. A plane tangent to the cylinder meets the cylinder in a straight line segment. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, a cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively. Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid. The volume of a cylinder with height h is V = ∫0 h A d x = ∫0 h π a b d x = π a b ∫0 h d x = π a b h. Even more general than the cylinder is the generalized cylinder. The cylinder is a degenerate quadric because at least one of the coordinates does not appear in the equation, an oblique cylinder has the top and bottom surfaces displaced from one another. There are other unusual types of cylinders. Let the height be h, internal radius r, and external radius R, the volume is given by V = π h
27.
Cone
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A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. A cone is formed by a set of segments, half-lines, or lines connecting a common point. If the enclosed points are included in the base, the cone is a solid object, otherwise it is a two-dimensional object in three-dimensional space. In the case of an object, the boundary formed by these lines or partial lines is called the lateral surface, if the lateral surface is unbounded. In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, in the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a cone on one side of the apex is called a nappe. The axis of a cone is the line, passing through the apex. If the base is right circular the intersection of a plane with this surface is a conic section, in general, however, the base may be any shape and the apex may lie anywhere. Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly, a cone with a polygonal base is called a pyramid. Depending on the context, cone may also mean specifically a convex cone or a projective cone, cones can also be generalized to higher dimensions. The perimeter of the base of a cone is called the directrix, the base radius of a circular cone is the radius of its base, often this is simply called the radius of the cone. The aperture of a circular cone is the maximum angle between two generatrix lines, if the generatrix makes an angle θ to the axis, the aperture is 2θ. A cone with a region including its apex cut off by a plane is called a cone, if the truncation plane is parallel to the cones base. An elliptical cone is a cone with an elliptical base, a generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary. The slant height of a circular cone is the distance from any point on the circle to the apex of the cone. It is given by r 2 + h 2, where r is the radius of the cirf the cone and this application is primarily useful in determining the slant height of a cone when given other information regarding the radius or height. The volume V of any conic solid is one third of the product of the area of the base A B and the height h V =13 A B h. In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral ∫ x 2 d x =13 x 3
28.
Pyramid (geometry)
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In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face and it is a conic solid with polygonal base. A pyramid with a base has n +1 vertices, n +1 faces. A right pyramid has its apex directly above the centroid of its base, nonright pyramids are called oblique pyramids. A regular pyramid has a polygon base and is usually implied to be a right pyramid. When unspecified, a pyramid is usually assumed to be a square pyramid. A triangle-based pyramid is often called a tetrahedron. Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called if its apex is above the interior of the base and obtuse if its apex is above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base, in a tetrahedron these qualifiers change based on which face is considered the base. Pyramids are a subclass of the prismatoids, pyramids can be doubled into bipyramids by adding a second offset point on the other side of the base plane. A right pyramid with a base has isosceles triangle sides, with symmetry is Cnv or. It can be given an extended Schläfli symbol ∨, representing a point, a join operation creates a new edge between all pairs of vertices of the two joined figures. The trigonal or triangular pyramid with all equilateral triangles faces becomes the regular tetrahedron, a lower symmetry case of the triangular pyramid is C3v, which has an equilateral triangle base, and 3 identical isosceles triangle sides. The square and pentagonal pyramids can also be composed of convex polygons. Right pyramids with regular star polygon bases are called star pyramids, for example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides. A right pyramid can be named as ∨P, where is the point, ∨ is a join operator. It has C1v symmetry from two different base-apex orientations, and C2v in its full symmetry, a rectangular right pyramid, written as ∨, and a rhombic pyramid, as ∨, both have symmetry C2v. The volume of a pyramid is V =13 b h and this works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane containing the base
29.
Square pyramid
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In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, if the sides are all equilateral triangles, the pyramid is one of the Johnson solids. The 92 Johnson solids were named and described by Norman Johnson in 1966, a Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform. They were named by Norman Johnson, who first listed these polyhedra in 1966, the Johnson square pyramid can be characterized by a single edge-length parameter a. The height H, the surface area A, and the volume V of such a pyramid are, other square pyramids have isosceles triangle sides. For square pyramids in general, with length l and height h. Square pyramids fill space with tetrahedra, truncated cubes or cuboctahedra, the square pyramid is topologically a self-dual polyhedron. The dual edge lengths are different due to the polar reciprocation, like all pyramids, the square pyramid is self-dual, having the same number of vertices as faces. A square pyramid can be represented by the Wheel graph W5, eric W. Weisstein, Square pyramid at MathWorld. Square Pyramid -- Interactive Polyhedron Model Virtual Reality Polyhedra www. georgehart. com, The Encyclopedia of Polyhedra
30.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
31.
Archimedes
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Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics and he is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Archimedes was born c.287 BC in the city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years, in The Sand Reckoner, Archimedes gives his fathers name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, a biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt and he referred to Conon of Samos as his friend, while two of his works have introductions addressed to Eratosthenes. Archimedes died c.212 BC during the Second Punic War, according to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, the soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives an account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable asset and had ordered that he not be harmed. Marcellus called Archimedes a geometrical Briareus, the last words attributed to Archimedes are Do not disturb my circles, a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is given in Latin as Noli turbare circulos meos. The phrase is given in Katharevousa Greek as μὴ μου τοὺς κύκλους τάραττε
32.
Accessible surface area
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The accessible surface area or solvent-accessible surface area is the surface area of a biomolecule that is accessible to a solvent. Measurement of ASA is usually described in units of square Ångstroms, ASA was first described by Lee & Richards in 1971 and is sometimes called the Lee-Richards molecular surface. ASA is typically calculated using the rolling ball algorithm developed by Shrake & Rupley in 1973 and this algorithm uses a sphere of a particular radius to probe the surface of the molecule. The points are drawn at a water molecules estimated radius beyond the van der Waals radius, all points are checked against the surface of neighboring atoms to determine whether they are buried or accessible. The number of points accessible is multiplied by the portion of area each point represents to calculate the ASA. The choice of the probe radius does have an effect on the surface area, as using a smaller probe radius detects more surface details. A typical value is 1. 4Å, which approximates the radius of a water molecule, another factor that affects the results is the definition of the VDW radii of the atoms in the molecule under study. For example, the molecule may often lack hydrogen atoms which are implicit in the structure, the hydrogen atoms may be implicitly included in the atomic radii of the heavy atoms, with a measure called the group radii. In addition, the number of points created on the van der Waals surface of each atom determines another aspect of discretization, the LCPO method uses a linear approximation of the two-body problem for a quicker analytical calculation of ASA. The approximations used in LCPO result in an error in the range of 1-3 Å², recently a method was presented that calculates ASA fast and analytically using a power diagram. Accessible surface area is used when calculating the transfer free energy required to move a biomolecule from aqueous solvent to a non-polar solvent such as a lipid environment. The LCPO method is used when calculating implicit solvent effects in the molecular dynamics software package AMBER. It is recently suggested that surface area can be used to improve prediction of protein secondary structure. The ASA is closely related to the concept of the solvent-excluded surface and it is also calculated in practice via a rolling-ball algorithm developed by Frederic Richards and implemented three-dimensionally by Michael Connolly in 1983 and Tim Richmond in 1984. Connolly spent several years perfecting the method. Richmond, Timothy J. solvent accessible surface area and excluded volume in proteins, Connolly, Michael L. Computation of molecular volume. Connolly, M. L. Molecular interstitial skeleton, modelling and Applications of Molecular Surfaces. Blaney, J. M. Distance Geometry in Molecular Modeling, grant, J. A. Pickup, B. T
33.
Chemical kinetics
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Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Van t Hoff studied chemical dynamics and published in 1884 his famous Etudes de dynamique chimique, after van t Hoff, chemical kinetics deals with the experimental determination of reaction rates from which rate laws and rate constants are derived. Relatively simple rate laws exist for zero order reactions, first order reactions, and second order reactions, and can be derived for others. Elementary reactions follow the law of action, but the rate law of stepwise reactions has to be derived by combining the rate laws of the various elementary steps. In consecutive reactions, the rate-determining step often determines the kinetics, in consecutive first order reactions, a steady state approximation can simplify the rate law. The activation energy for a reaction is determined through the Arrhenius equation. Gorban and Yablonsky have suggested that the history of chemical dynamics can be divided into three eras, the first is the van t Hoff wave searching for the general laws of chemical reactions and relating kinetics to thermodynamics. The second may be called the Semenov--Hinshelwood wave with emphasis on reaction mechanisms, the third is associated with Aris and the detailed mathematical description of chemical reaction networks. Depending upon what substances are reacting, the rate varies. Acid/base reactions, the formation of salts, and ion exchange are fast reactions, when covalent bond formation takes place between the molecules and when large molecules are formed, the reactions tend to be very slow. Nature and strength of bonds in reactant molecules greatly influence the rate of its transformation into products, the physical state of a reactant is also an important factor of the rate of change. When reactants are in the phase, as in aqueous solution. However, when they are in different phases, the reaction is limited to the interface between the reactants, reaction can occur only at their area of contact, in the case of a liquid and a gas, at the surface of the liquid. Vigorous shaking and stirring may be needed to bring the reaction to completion and this means that the more finely divided a solid or liquid reactant the greater its surface area per unit volume and the more contact it with the other reactant, thus the faster the reaction. To make an analogy, for example, when one starts a fire, one uses wood chips, in organic chemistry, on water reactions are the exception to the rule that homogeneous reactions take place faster than heterogeneous reactions. In a solid, only those particles that are at the surface can be involved in a reaction, for example, Sherbet is a mixture of very fine powder of malic acid and sodium hydrogen carbonate. On contact with the saliva in the mouth, these chemicals quickly dissolve and react, releasing carbon dioxide, also, fireworks manufacturers modify the surface area of solid reactants to control the rate at which the fuels in fireworks are oxidised, using this to create different effects. For example, finely divided aluminium confined in a shell explodes violently, if larger pieces of aluminium are used, the reaction is slower and sparks are seen as pieces of burning metal are ejected
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Reaction rate
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The reaction rate or speed of reaction for a reactant or product in a particular reaction is intuitively defined as how quickly or slowly a reaction takes place. Chemical kinetics is the part of chemistry that studies reaction rates. The concepts of chemical kinetics are applied in many disciplines, such as engineering, enzymology. Consider a typical reaction, a A + b B → p P + q Q The lowercase letters represent stoichiometric coefficients, while the capital letters represent the reactants. Reaction rate usually has the units of mol L−1 s−1, the rate of a reaction is always positive. A negative sign is present to indicate that the reactant concentration is decreasing. )The IUPAC recommends that the unit of time should always be the second. The rate of reaction differs from the rate of increase of concentration of a product P by a constant factor and for a reactant A by minus the reciprocal of the stoichiometric number. The stoichiometric numbers are included so that the rate is independent of which reactant or product species is chosen for measurement. For example, if a =1 and b =3 then B is consumed three times more rapidly than A, but v = -d/dt = -d/dt is uniquely defined. The above definition is valid for a single reaction, in a closed system of constant volume. If water is added to a pot containing salty water, the concentration of salt decreases, although there is no chemical reaction. When applied to the system at constant volume considered previously, this equation reduces to, r = d d t. Here N0 is the Avogadro constant, for a single reaction in a closed system of varying volume the so-called rate of conversion can be used, in order to avoid handling concentrations. It is defined as the derivative of the extent of reaction with respect to time, also V is the volume of reaction and Ci is the concentration of substance i. When side products or reaction intermediates are formed, the IUPAC recommends the use of the rate of appearance and rate of disappearance for products and reactants. Reaction rates may also be defined on a basis that is not the volume of the reactor, when a catalyst is used the reaction rate may be stated on a catalyst weight or surface area basis. If the basis is a specific catalyst site that may be counted by a specified method. The nature of the reaction, Some reactions are faster than others
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Chemical reaction
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A chemical reaction is a process that leads to the transformation of one set of chemical substances to another. Nuclear chemistry is a sub-discipline of chemistry that involves the reactions of unstable. The substance initially involved in a reaction are called reactants or reagents. Chemical reactions are characterized by a chemical change, and they yield one or more products. Reactions often consist of a sequence of individual sub-steps, the elementary reactions. Chemical reactions are described with chemical equations, which present the starting materials, end products. Chemical reactions happen at a characteristic reaction rate at a given temperature, typically, reaction rates increase with increasing temperature because there is more thermal energy available to reach the activation energy necessary for breaking bonds between atoms. Reactions may proceed in the forward or reverse direction until they go to completion or reach equilibrium, Reactions that proceed in the forward direction to approach equilibrium are often described as spontaneous, requiring no input of free energy to go forward. Non-spontaneous reactions require input of energy to go forward. Different chemical reactions are used in combinations during chemical synthesis in order to obtain a desired product, in biochemistry, a consecutive series of chemical reactions form metabolic pathways. These reactions are catalyzed by protein enzymes. Chemical reactions such as combustion in fire, fermentation and the reduction of ores to metals were known since antiquity, in the Middle Ages, chemical transformations were studied by Alchemists. They attempted, in particular, to lead into gold, for which purpose they used reactions of lead. The process involved heating of sulfate and nitrate minerals such as sulfate, alum. In the 17th century, Johann Rudolph Glauber produced hydrochloric acid and sodium sulfate by reacting sulfuric acid, further optimization of sulfuric acid technology resulted in the contact process in the 1880s, and the Haber process was developed in 1909–1910 for ammonia synthesis. From the 16th century, researchers including Jan Baptist van Helmont, Robert Boyle, the phlogiston theory was proposed in 1667 by Johann Joachim Becher. It postulated the existence of an element called phlogiston, which was contained within combustible bodies. This proved to be false in 1785 by Antoine Lavoisier who found the explanation of the combustion as reaction with oxygen from the air
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Iron
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Iron is a chemical element with symbol Fe and atomic number 26. It is a metal in the first transition series and it is by mass the most common element on Earth, forming much of Earths outer and inner core. It is the fourth most common element in the Earths crust, like the other group 8 elements, ruthenium and osmium, iron exists in a wide range of oxidation states, −2 to +6, although +2 and +3 are the most common. Elemental iron occurs in meteoroids and other low oxygen environments, but is reactive to oxygen, fresh iron surfaces appear lustrous silvery-gray, but oxidize in normal air to give hydrated iron oxides, commonly known as rust. Unlike the metals that form passivating oxide layers, iron oxides occupy more volume than the metal and thus flake off, Iron metal has been used since ancient times, although copper alloys, which have lower melting temperatures, were used even earlier in human history. Pure iron is soft, but is unobtainable by smelting because it is significantly hardened and strengthened by impurities, in particular carbon. A certain proportion of carbon steel, which may be up to 1000 times harder than pure iron. Crude iron metal is produced in blast furnaces, where ore is reduced by coke to pig iron, further refinement with oxygen reduces the carbon content to the correct proportion to make steel. Steels and iron alloys formed with metals are by far the most common industrial metals because they have a great range of desirable properties. Iron chemical compounds have many uses, Iron oxide mixed with aluminium powder can be ignited to create a thermite reaction, used in welding and purifying ores. Iron forms binary compounds with the halogens and the chalcogens, among its organometallic compounds is ferrocene, the first sandwich compound discovered. Iron plays an important role in biology, forming complexes with oxygen in hemoglobin and myoglobin. Iron is also the metal at the site of many important redox enzymes dealing with cellular respiration and oxidation and reduction in plants. A human male of average height has about 4 grams of iron in his body and this iron is distributed throughout the body in hemoglobin, tissues, muscles, bone marrow, blood proteins, enzymes, ferritin, hemosiderin, and transport in plasma. The mechanical properties of iron and its alloys can be evaluated using a variety of tests, including the Brinell test, Rockwell test, the data on iron is so consistent that it is often used to calibrate measurements or to compare tests. An increase in the content will cause a significant increase in the hardness. Maximum hardness of 65 Rc is achieved with a 0. 6% carbon content, because of the softness of iron, it is much easier to work with than its heavier congeners ruthenium and osmium. Because of its significance for planetary cores, the properties of iron at high pressures and temperatures have also been studied extensively
37.
Combustion
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Combustion in a fire produces a flame, and the heat produced can make combustion self-sustaining. Combustion is often a sequence of elementary radical reactions. Solid fuels, such as wood, first undergo endothermic pyrolysis to produce gaseous fuels whose combustion then supplies the required to produce more of them. Combustion is often hot enough that light in the form of either glowing or a flame is produced, a simple example can be seen in the combustion of hydrogen and oxygen into water vapor, a reaction commonly used to fuel rocket engines. The bond energies in the play only a minor role, since they are similar to those in the combustion products. The heat of combustion is approximately -418 kJ per mole of O2 used up in the combustion reaction, uncatalyzed combustion in air requires fairly high temperatures. Complete combustion is stoichiometric with respect to the fuel, where there is no remaining fuel, thermodynamically, the chemical equilibrium of combustion in air is overwhelmingly on the side of the products. Thus, the smoke is usually toxic and contains unburned or partially oxidized products. Since combustion is rarely clean, flue gas cleaning or catalytic converters may be required by law, fires occur naturally, ignited by lightning strikes or by volcanic products. Combustion was the first controlled chemical reaction discovered by humans, in the form of campfires and bonfires, usually, the fuel is carbon, hydrocarbons or more complicated mixtures such as wood that contains partially oxidized hydrocarbons. Combustion is also currently the only used to power rockets. Combustion is also used to destroy waste, both nonhazardous and hazardous, oxidants for combustion have high oxidation potential and include atmospheric or pure oxygen, chlorine, fluorine, chlorine trifluoride, nitrous oxide and nitric acid. For instance, hydrogen burns in chlorine to form hydrogen chloride with the liberation of heat, although usually not catalyzed, combustion can be catalyzed by platinum or vanadium, as in the contact process. In complete combustion, the reactant burns in oxygen, producing a number of products. When a hydrocarbon burns in oxygen, the reaction will yield carbon dioxide. When elements are burned, the products are primarily the most common oxides, carbon will yield carbon dioxide, sulfur will yield sulfur dioxide, and iron will yield iron oxide. Nitrogen is not considered to be a combustible substance when oxygen is the oxidant, Combustion is not necessarily favorable to the maximum degree of oxidation, and it can be temperature-dependent. For example, sulfur trioxide is not produced quantitatively by the combustion of sulfur, NOx species appear in significant amounts above about 2,800 °F, and more is produced at higher temperatures
38.
Surface-area-to-volume ratio
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For a given volume, the object with the smallest surface area is the sphere, a consequence of the isoperimetric inequality in 3 dimensions. By contrast, objects with tiny spikes will have large surface area for a given volume. If a cell is too large, not enough materials will be able to cross the membrane fast enough to accommodate the high cellular volume, the surface-area-to-volume ratio has physical dimension L−1 and is therefore expressed in units of inverse distance. As an example, a cube with sides of length 1 cm will have an area of 6 cm2. The surface to volume ratio for this cube is thus SA, for a given shape, SA, V is inversely proportional to size. A cube 2 cm on a side has a ratio of 3 cm−1, conversely, preserving SA, V as size increases requires changing to a less compact shape. Materials with high surface area to volume ratio react at much faster rates than monolithic materials, examples include grain dust, while grain isnt typically flammable, grain dust is explosive. Finely ground salt dissolves much more quickly than coarse salt, high surface area to volume ratio provides a strong driving force to speed up thermodynamic processes that minimize free energy. The ratio between the area and volume of cells and organisms has an enormous impact on their biology. For example, many microorganisms have increased surface area to increase their drag in the water. This reduces their rate of sink and allows them to remain near the surface with less energy expenditure, an increased surface area to volume ratio also means increased exposure to the environment. The many tentacles of jellyfish and anemones are the result of increased area for the acquisition of food. Greater surface area allows more of the water to be sifted for food. Individual organs in animals are based on the principle of greater surface area. The lung is an organ with numerous internal branchings that increase the area through which oxygen is passed into the blood. The intestine has a finely wrinkled surface, increasing the area through which nutrients are absorbed by the body. This is done to increase the area in which diffusion of oxygen and carbon dioxide in the lungs. Cells can achieve a high area to volume ratio by being long
39.
Mitochondrion
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The mitochondrion is a double membrane-bound organelle found in all eukaryotic organisms. Some cells in multicellular organisms may however lack them. A number of organisms, such as microsporidia, parabasalids. To date, only one eukaryote, Monocercomonoides, is known to have completely lost its mitochondria, the word mitochondrion comes from the Greek μίτος, mitos, thread, and χονδρίον, chondrion, granule or grain-like. Mitochondria generate most of the supply of adenosine triphosphate, used as a source of chemical energy. Mitochondria are commonly between 0.75 and 3 μm in diameter but vary considerably in size and structure, unless specifically stained, they are not visible. Mitochondrial biogenesis is in turn temporally coordinated with these cellular processes, Mitochondria have been implicated in several human diseases, including mitochondrial disorders, cardiac dysfunction, heart failure and autism. The number of mitochondria in a cell can vary widely by organism, tissue, for instance, red blood cells have no mitochondria, whereas liver cells can have more than 2000, The organelle is composed of compartments that carry out specialized functions. These compartments or regions include the outer membrane, the space, the inner membrane. Although most of a cells DNA is contained in the cell nucleus, mitochondrial proteins vary depending on the tissue and the species. In humans,615 distinct types of protein have been identified from cardiac mitochondria, the mitochondrial proteome is thought to be dynamically regulated. The first observations of structures that probably represented mitochondria were published in the 1840s. Richard Altmann, in 1890, established them as cell organelles, the term mitochondria was coined by Carl Benda in 1898. Leonor Michaelis discovered that Janus green can be used as a stain for mitochondria in 1900. Benjamin F. Kingsbury, in 1912, first related them with cell respiration, in 1913, particles from extracts of guinea-pig liver were linked to respiration by Otto Heinrich Warburg, which he called grana. Warburg and Heinrich Otto Wieland, who had postulated a similar particle mechanism. It was not until 1925, when David Keilin discovered cytochromes, in the following years, the mechanism behind cellular respiration was further elaborated, although its link to the mitochondria was not known. The introduction of tissue fractionation by Albert Claude allowed mitochondria to be isolated from other cell fractions, in 1946, he concluded that cytochrome oxidase and other enzymes responsible for the respiratory chain were isolated to the mitchondria
40.
Cellular respiration
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Respiration is one of the key ways a cell releases chemical energy to fuel cellular activity. Cellular respiration is considered an exothermic reaction which releases heat. The overall reaction occurs in a series of steps, most of which are redox reactions themselves. Nutrients that are used by animal and plant cells in respiration include sugar, amino acids and fatty acids. The chemical energy stored in ATP can then be used to drive processes requiring energy, including biosynthesis, aerobic respiration requires oxygen in order to create ATP. The potential of NADH and FADH2 is converted to more ATP through a transport chain with oxygen as the terminal electron acceptor. Most of the ATP produced by cellular respiration is made by oxidative phosphorylation. This works by the released in the consumption of pyruvate being used to create a chemiosmotic potential by pumping protons across a membrane. This potential is used to drive ATP synthase and produce ATP from ADP. Biology textbooks often state that 38 ATP molecules can be made per oxidised glucose molecule during cellular respiration, aerobic metabolism is up to 15 times more efficient than anaerobic metabolism. They share the pathway of glycolysis but aerobic metabolism continues with the Krebs cycle. The post-glycolytic reactions take place in the mitochondria in eukaryotic cells, glycolysis is a metabolic pathway that takes place in the cytosol of cells in all living organisms. This pathway can function with or without the presence of oxygen, in humans, aerobic conditions produce pyruvate and anaerobic conditions produce lactate. In aerobic conditions, the process converts one molecule of glucose into two molecules of pyruvate, generating energy in the form of two net molecules of ATP, four molecules of ATP per glucose are actually produced, however, two are consumed as part of the preparatory phase. The initial phosphorylation of glucose is required to increase the reactivity in order for the molecule to be cleaved into two molecules by the enzyme aldolase. During the pay-off phase of glycolysis, four groups are transferred to ADP by substrate-level phosphorylation to make four ATP. Glycogen can be converted into glucose 6-phosphate as well with the help of glycogen phosphorylase, during energy metabolism, glucose 6-phosphate becomes fructose 6-phosphate. An additional ATP is used to phosphorylate fructose 6-phosphate into fructose 1, fructose 1, 6-diphosphate then splits into two phosphorylated molecules with three carbon chains which later degrades into pyruvate
41.
Micrograph
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A micrograph or photomicrograph is a photograph or digital image taken through a microscope or similar device to show a magnified image of an item. This is opposed to an image, which is at a scale that is visible to the naked eye. Micrography is the practice or art of using microscopes to make photographs, a micrograph contains extensive details that form the features of a microstructure. The neuropathologist Solomon C. Fuller designed and created the first photomicrograph in 1900, micrographs are widely used in all fields of microscopy. A light micrograph or photomicrograph is a micrograph prepared using an optical microscope, at a basic level, photomicroscopy may be performed simply by hooking up a regular camera to a microscope, thereby enabling the user to take photographs at reasonably high magnification. Roman Vishniac was a pioneer in the field of photomicroscopy, specializing in the photography of living creatures in full motion and he also made major developments in light-interruption photography and color photomicroscopy. An electron micrograph is a micrograph prepared using an electron microscope, however, the term electron micrograph is not used in electron microscopy. Digital micrography is a digital picture obtained either directly with a microscope or by scanning of a photomicrograph, digital micrographs are now commonly obtained using a USB microscope attached directly to a home computer or laptop. Today, an add-on three-in-one macro lens which has capability to take wide-angle, fish-eye and macro with 7x, 14x, micrographs usually have micron bars, or magnification ratios, or both. Magnification is a ratio between size of object on a picture and its real size, unfortunately, magnification is somewhat a misleading parameter. It depends on a size of a printed picture. Editors of journals and magazines routinely resize a figure to fit the page, a scale bar, or micron bar, is a bar of known length displayed on a picture. The bar can be used for measurements on a picture, when a picture is resized a bar is also resized. If a picture has a bar, the magnification can be easily calculated, ideally, all pictures destined for publication/presentation should be supplied with a scale bar, the magnification ratio is optional. All but one of the micrographs presented on this page do not have a bar, supplied magnification ratios are likely incorrect. The microscope has been used for scientific discovery. It has also linked to the arts since its invention in the 17th century. At first scientists used the microscope to view and draw objects not visible with the unaided eye, early adopters of the microscope, such as Robert Hooke and Antonie van Leeuwenhoek, were excellent illustrators
42.
Digestion
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Digestion is the breakdown of large insoluble food molecules into small water-soluble food molecules so that they can be absorbed into the watery blood plasma. In certain organisms, these substances are absorbed through the small intestine into the blood stream. Digestion is a form of catabolism that is divided into two processes based on how food is broken down, mechanical and chemical digestion. The term mechanical digestion refers to the breakdown of large pieces of food into smaller pieces which can subsequently be accessed by digestive enzymes. In chemical digestion, enzymes break down food into the small molecules the body can use. In the human system, food enters the mouth and mechanical digestion of the food starts by the action of mastication, a form of mechanical digestion. After undergoing mastication and starch digestion, the food will be in the form of a small and it will then travel down the esophagus and into the stomach by the action of peristalsis. Gastric juice in the stomach starts protein digestion, gastric juice mainly contains hydrochloric acid and pepsin. As these two chemicals may damage the stomach wall, mucus is secreted by the stomach, providing a slimy layer that acts as a shield against the effects of the chemicals. At the same time protein digestion is occurring, mechanical mixing occurs by peristalsis and this allows the mass of food to further mix with the digestive enzymes. After some time, the resulting liquid is called chyme. When the chyme is fully digested, it is absorbed into the blood, 95% of absorption of nutrients occurs in the small intestine. Water and minerals are reabsorbed back into the blood in the colon where the pH is slightly acidic about 5.6 ~6.9, some vitamins, such as biotin and vitamin K produced by bacteria in the colon are also absorbed into the blood in the colon. Waste material is eliminated from the rectum during defecation, there is a fundamental distinction between internal and external digestion. External digestion developed earlier in history, and most fungi still rely on it. In this process, enzymes are secreted into the environment surrounding the organism, where they break down an organic material, some organisms, including nearly all spiders, simply secrete biotoxins and digestive chemicals into the extracellular environment prior to ingestion of the consequent soup. Bacteria use several systems to obtain nutrients from other organisms in the environments, in a channel transupport system, several proteins form a contiguous channel traversing the inner and outer membranes of the bacteria. It is a system, which consists of only three protein subunits, the ABC protein, membrane fusion protein, and outer membrane protein
43.
Tooth
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A tooth is a hard, calcified structure found in the jaws of many vertebrates and used to break down food. Some animals, particularly carnivores, also use teeth for hunting or for defensive purposes, the roots of teeth are covered by gums. Teeth are not made of bone, but rather of tissues of varying density. The cellular tissues that ultimately become teeth originate from the germ layer. The general structure of teeth is similar across the vertebrates, although there is variation in their form. The teeth of mammals have deep roots, and this pattern is found in some fish. In most teleost fish, however, the teeth are attached to the surface of the bone. In cartilaginous fish, such as sharks, the teeth are attached by tough ligaments to the hoops of cartilage that form the jaw, some animals develop only one set of teeth while others develop many sets. Sharks, for example, grow a new set of every two weeks to replace worn teeth. Rodent incisors grow and wear away continually through gnawing, which helps maintain relatively constant length, the industry of the beaver is due in part to this qualification. Many rodents such as voles and guinea pigs, but not mice, Teeth are not always attached to the jaw, as they are in mammals. In many reptiles and fish, teeth are attached to the palate or to the floor of the mouth, some teleosts even have teeth in the pharynx. While not true teeth in the sense, the dermal denticles of sharks are almost identical in structure and are likely to have the same evolutionary origin. Though modern teeth-like structures with dentine and enamel have been found in late conodonts, living amphibians typically have small teeth, or none at all, since they commonly feed only on soft foods. In reptiles, teeth are simple and conical in shape. The pattern of incisors, canines, premolars and molars is found only in mammals, the numbers of these types of teeth vary greatly between species, zoologists use a standardised dental formula to describe the precise pattern in any given group. The genes governing tooth development in mammals are homologous to these involved in the development of fish scales, Teeth are among the most distinctive features of mammal species. Paleontologists use teeth to identify species and determine their relationships
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Microvillus
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Thousands of microvilli form a structure called the brush border that is found on the apical surface of some epithelial cells, such as the small intestines. Microvilli are observed on the surface of eggs, aiding in the anchoring of sperm cells that have penetrated the extracellular coat of egg cells. Clustering of elongated microtubules around a sperm allows for it to be drawn closer and they are large objects that increase surface area for absorption. Microvilli are also of importance on the surface of white blood cells. Microvilli are covered in plasma membrane, which encloses cytoplasm and microfilaments, though these are cellular extensions, there are little or no cellular organelles present in the microvilli. Each microvillus has a bundle of cross-linked actin filaments, which serves as its structural core. 20 to 30 tightly bundled actin filaments are cross-linked by bundling proteins fimbrin, villin and espin to form the core of the microvilli. In the enterocyte microvillus, the core is attached to the plasma membrane along its length by lateral arms made of myosin 1a. Myosin 1a functions through a site for filamentous actin on one end. As mentioned, microvilli are formed as cell extensions from the membrane surface. Actin filaments, present in the cytosol, are most abundant near the cell surface and these filaments are thought to determine the shape and movement of the plasma membrane. The nucleation of actin fibers occurs as a response to external stimuli and this could account for the uniformity of the microvilli, which are observed to be of equal length and diameter. This nucleation process occurs from the end, allowing rapid growth from the plus end. Though the length and composition of microvilli is consistent within a group of homogenous cells. For example, the microvilli in the small and large intestines in mice are different in length. Microvilli function as the surface of nutrient absorption in the gastrointestinal tract. Because of this function, the microvillar membrane is packed with enzymes that aid in the breakdown of complex nutrients into simpler compounds that are more easily absorbed. For example, enzymes that digest carbohydrates called glycosidases are present at concentrations on the surface of enterocyte microvilli