A solar eclipse occurs when an observer passes through the shadow cast by the Moon which or blocks the Sun. This can only happen when the Sun and Earth are nearly aligned on a straight line in three dimensions during a new moon when the Moon is close to the ecliptic plane. In a total eclipse, the disk of the Sun is obscured by the Moon. In partial and annular eclipses, only part of the Sun is obscured. If the Moon were in a circular orbit, a little closer to the Earth, in the same orbital plane, there would be total solar eclipses every new moon. However, since the Moon's orbit is tilted at more than 5 degrees to the Earth's orbit around the Sun, its shadow misses Earth. A solar eclipse can only occur when the moon is close enough to the ecliptic plane during a new moon. Special conditions must occur for the two events to coincide because the Moon's orbit crosses the ecliptic at its orbital nodes twice every draconic month while a new moon occurs one every synodic month. Solar eclipses therefore happen only during eclipse seasons resulting in at least two, up to five, solar eclipses each year.
Total eclipses are rare because the timing of the new moon within the eclipse season needs to be more exact for an alignment between the observer and the centers of the Sun and Moon. In addition, the elliptical orbit of the Moon takes it far enough away from Earth that its apparent size is not large enough to block the Sun entirely. Total solar eclipses are rare at any particular location because totality exists only along a narrow path on the Earth's surface traced by the Moon's full shadow or umbra. An eclipse is a natural phenomenon. However, in some ancient and modern cultures, solar eclipses were attributed to supernatural causes or regarded as bad omens. A total solar eclipse can be frightening to people who are unaware of its astronomical explanation, as the Sun seems to disappear during the day and the sky darkens in a matter of minutes. Since looking directly at the Sun can lead to permanent eye damage or blindness, special eye protection or indirect viewing techniques are used when viewing a solar eclipse.
It is technically safe to view only the total phase of a total solar eclipse with the unaided eye and without protection. People referred to as eclipse chasers or umbraphiles will travel to remote locations to observe or witness predicted central solar eclipses. There are four types of solar eclipses: A total eclipse occurs when the dark silhouette of the Moon obscures the intensely bright light of the Sun, allowing the much fainter solar corona to be visible. During any one eclipse, totality occurs at best only in a narrow track on the surface of Earth; this narrow track is called the path of totality. An annular eclipse occurs when the Sun and Moon are in line with the Earth, but the apparent size of the Moon is smaller than that of the Sun. Hence the Sun appears as a bright ring, or annulus, surrounding the dark disk of the Moon. A hybrid eclipse shifts between a annular eclipse. At certain points on the surface of Earth, it appears as a total eclipse, whereas at other points it appears as annular.
Hybrid eclipses are comparatively rare. A partial eclipse occurs when the Sun and Moon are not in line with the Earth and the Moon only obscures the Sun; this phenomenon can be seen from a large part of the Earth outside of the track of an annular or total eclipse. However, some eclipses can only be seen as a partial eclipse, because the umbra passes above the Earth's polar regions and never intersects the Earth's surface. Partial eclipses are unnoticeable in terms of the sun's brightness, as it takes well over 90% coverage to notice any darkening at all. At 99%, it would be no darker than civil twilight. Of course, partial eclipses can be observed; the Sun's distance from Earth is about 400 times the Moon's distance, the Sun's diameter is about 400 times the Moon's diameter. Because these ratios are the same, the Sun and the Moon as seen from Earth appear to be the same size: about 0.5 degree of arc in angular measure. A separate category of solar eclipses is that of the Sun being occluded by a body other than the Earth's moon, as can be observed at points in space away from the Earth's surface.
Two examples are when the crew of Apollo 12 observed the Earth eclipse the Sun in 1969 and when the Cassini probe observed Saturn eclipsing the Sun in 2006. The Moon's orbit around the Earth is elliptical, as is the Earth's orbit around the Sun; the apparent sizes of the Sun and Moon therefore vary. The magnitude of an eclipse is the ratio of the apparent size of the Moon to the apparent size of the Sun during an eclipse. An eclipse that occurs when the Moon is near its closest distance to Earth can be a total eclipse because the Moon will appear to be large enough to cover the Sun's bright disk or photosphere. Conversely, an eclipse that occurs when the Moon is near its farthest distance from Earth can only be an annular eclipse because the Moon will appear to be smaller than the Sun. More solar eclipses are
In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope though these entities were quantitatively small; the word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object, smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it must be compared to another infinitesimal object in the same context. Infinitely many infinitesimals are summed to produce an integral.
The concept of infinitesimals was introduced around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz. Archimedes used what came to be known as the method of indivisibles in his work The Method of Mechanical Theorems to find areas of regions and volumes of solids. In his formal published treatises, Archimedes solved the same problem using the method of exhaustion; the 15th century saw the work of Nicholas of Cusa, further developed in the 17th century by Johannes Kepler, in particular calculation of area of a circle by representing the latter as an infinite-sided polygon. Simon Stevin's work on decimal representation of all numbers in the 16th century prepared the ground for the real continuum. Bonaventura Cavalieri's method of indivisibles led to an extension of the results of the classical authors; the method of indivisibles related to geometrical figures as being composed of entities of codimension 1. John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures into infinitely thin building blocks of the same dimension as the figure, preparing the ground for general methods of the integral calculus.
He exploited an infinitesimal denoted 1/∞ in area calculations. The use of infinitesimals by Leibniz relied upon heuristic principles, such as the law of continuity: what succeeds for the finite numbers succeeds for the infinite numbers and vice versa; the 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange. Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, in defining an early form of a Dirac delta function; as Cantor and Dedekind were developing more abstract versions of Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions. Du Bois-Reymond's work inspired both Émile Thoralf Skolem. Borel explicitly linked du Bois-Reymond's work to Cauchy's work on rates of growth of infinitesimals. Skolem developed the first non-standard models of arithmetic in 1934. A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed non-standard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955.
The hyperreals implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity. The standard part function implements Fermat's adequality. Vladimir Arnold wrote in 1990: Nowadays, when teaching analysis, it is not popular to talk about infinitesimal quantities. Present-day students are not in command of this language, it is still necessary to have command of it. The notion of infinitely small quantities was discussed by the Eleatic School; the Greek mathematician Archimedes, in The Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinitesimals. His Archimedean property defines a number x as infinite if it satisfies the conditions |x|>1, |x|>1+1, |x|>1+1+1... and infinitesimal if x≠0 and a similar set of conditions holds for x and the reciprocals of the positive integers. A number system is said to be Archimedean if it contains infinitesimal members; the English mathematician John Wallis introduced the expression 1/∞ in his 1655 book Treatise on the Conic Sections.
The symbol, which denotes the reciprocal, or inverse, of ∞, is the symbolic representation of the mathematical concept of an infinitesimal. In his Treatise on the Conic Sections Wallis discusses the concept of a relationship between the symbolic representation of infinitesimal 1/∞ that he introduced and the concept of infinity for which he introduced the symbol ∞; the concept suggests a thought experiment of adding an infinite number of parallelograms of infinitesimal width to form a finite area. This concept was the predecessor to the modern method of integration used in integral calculus; the conceptual origins of the concept of the infinitesimal 1/∞ can be traced as far back as the Greek philosopher Zeno of Elea, whose Zeno's dichotomy paradox was the first mathematical concept to consider the relationship between a finite interval and an interval approaching that of an infinitesimal-sized interval. Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632.
Prior to the invention of calculus mathematicians were able to calculate tangent lines using Pierre de
In mathematics, the Pythagorean theorem known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides; the theorem can be written as an equation relating the lengths of the sides a, b and c called the "Pythagorean equation": a 2 + b 2 = c 2, where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. Although it is argued that knowledge of the theorem predates him, the theorem is named after the ancient Greek mathematician Pythagoras as it is he who, by tradition, is credited with its first proof, although no evidence of it exists. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework. Mesopotamian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases.
The theorem has been given numerous proofs – the most for any mathematical theorem. They are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years; the theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; the Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it. In any event, the proof attributed to him is simple, is called a proof by rearrangement; the two large squares shown in the figure each contain four identical triangles, the only difference between the two large squares is that the triangles are arranged differently. Therefore, the white space within each of the two large squares must have equal area.
Equating the area of the white space yields the Pythagorean theorem, Q. E. D; that Pythagoras originated this simple proof is sometimes inferred from the writings of the Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below. If c denotes the length of the hypotenuse and a and b denote the lengths of the other two sides, the Pythagorean theorem can be expressed as the Pythagorean equation: a 2 + b 2 = c 2. If the length of both a and b are known c can be calculated as c = a 2 + b 2. If the length of the hypotenuse c and of one side are known the length of the other side can be calculated as a = c 2 − b 2 or b = c 2 − a 2; the Pythagorean equation relates the sides of a right triangle in a simple way, so that if the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides, but less than their sum. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them.
If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other; this proof is based on the proportionality of the sides of two similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles. Let ABC represent a right triangle, with the right angle located at C. Draw the altitude from point C, call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e; the new triangle ACH is similar to triangle ABC, because they both have a right angle, they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as θ in the figure. By a similar reasoning, the triangle CBH is similar to ABC; the proof of similarity of the triangles requires the triangle postulate: the sum of the angles in a triangle is two right angles, is equivalent to the parallel postulate.
Similarity of the triangles leads to the equality of ratios of corresponding sides: B C A B = B H B C and A C A B = A H A C. The first result equates
A Möbius strip, Möbius band, or Möbius loop spelled Mobius or Moebius, is a surface with only one side and only one boundary. The Möbius strip has the mathematical property of being unorientable, it can be realized as a ruled surface. Its discovery is attributed to the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858, though a structure similar to the Möbius strip can be seen in Roman mosaics dated circa 200–250 AD. An example of a Möbius strip can be created by taking a paper strip and giving it a half-twist, joining the ends of the strip to form a loop. However, the Möbius strip is not a surface of only one exact size and shape, such as the half-twisted paper strip depicted in the illustration. Rather, mathematicians refer to the closed Möbius band as any surface, homeomorphic to this strip, its boundary is a simple closed curve, i.e. homeomorphic to a circle. This allows for a wide variety of geometric versions of the Möbius band as surfaces each having a definite size and shape.
For example, any rectangle can be glued to itself to make a Möbius band. Some of these can be smoothly modeled in Euclidean space, others cannot. A half-twist clockwise gives an embedding of the Möbius strip different from that of a half-twist counterclockwise – that is, as an embedded object in Euclidean space, the Möbius strip is a chiral object with right- or left-handedness. However, the underlying topological spaces within the Möbius strip are homeomorphic in each case. An infinite number of topologically different embeddings of the same topological space into three-dimensional space exist, as the Möbius strip can be formed by twisting the strip an odd number of times greater than one, or by knotting and twisting the strip, before joining its ends; the complete open Möbius band is an example of a topological surface, related to the standard Möbius strip, but, not homeomorphic to it. Finding algebraic equations, the solutions of which have the topology of a Möbius strip, is straightforward, but, in general, these equations do not describe the same geometric shape that one gets from the twisted paper model described above.
In particular, the twisted paper model is a developable surface. A system of differential-algebraic equations that describes models of this type was published in 2007 together with its numerical solution; the Euler characteristic of the Möbius strip is zero. The Möbius strip has several curious properties. A line drawn starting from the seam at the other side. If continued, the line meets the starting point, is double the length of the original strip; this single continuous curve demonstrates. Cutting a Möbius strip along the center line with a pair of scissors yields one long strip with two full twists in it, rather than two separate strips; this happens because the original strip only has one edge, twice as long as the original strip. Cutting creates a second independent edge, half of, on each side of the scissors. Cutting this new, strip down the middle creates two strips wound around each other, each with two full twists. If the strip is cut along about a third of the way in from the edge, it creates two strips: One is a thinner Möbius strip – it is the center third of the original strip, comprising one-third of the width and the same length as the original strip.
The other is a longer but thin strip with two full twists in it – this is a neighborhood of the edge of the original strip, it comprises one-third of the width and twice the length of the original strip. Other analogous strips can be obtained by joining strips with two or more half-twists in them instead of one. For example, a strip with three half-twists, when divided lengthwise, becomes a twisted strip tied in a trefoil knot. A strip with N half-twists, when bisected, becomes a strip with N + 1 full twists. Giving it extra twists and reconnecting the ends produces figures called paradromic rings. One way to represent the Möbius strip as a subset of three-dimensional Euclidean space is using the parametrization: x = cos u y = sin u z = v 2 sin u 2 where 0 ≤ u < 2 π and − 1 ≤ v ≤ 1. This creates a Möbius strip of width 1 whose center circle has radius 1, lies in the x y -plane and is centered at; the parameter u runs around th
The Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane, its name arises because it was discovered by Jakob Steiner when he was in Rome in 1844. The simplest construction is as the image of a sphere centered at the origin under the map f =; this gives an implicit formula of x 2 y 2 + y 2 z 2 + z 2 x 2 − r 2 x y z = 0. Taking a parametrization of the sphere in terms of longitude and latitude, gives parametric equations for the Roman surface as follows: x = r2 cos θ cos φ sin φ y = r2 sin θ cos φ sin φ z = r2 cos θ sin θ cos2 φ; the origin is a triple point, each of the xy-, yz-, xz-planes are tangential to the surface there. The other places of self-intersection are double points, defining segments along each coordinate axis which terminate in six pinch points; the entire surface has tetrahedral symmetry. It is a particular type of Steiner surface, that is, a 3-dimensional linear projection of the Veronese surface.
For simplicity we consider only the case r = 1. Given the sphere defined by the points such that x 2 + y 2 + z 2 = 1, we apply to these points the transformation T defined by T = =, say, but we have U 2 V 2 + V 2 W 2 + W 2 U 2 = z 2 x 2 y 4 + x 2 y 2 z 4 + y 2 z 2 x 4 = = = = U V W, so U 2 V 2 + V 2 W 2 + W 2 U 2 − U V W = 0 as desired. Conversely, suppose we are given satisfying U 2 V 2 + V 2 W 2 + W 2 U 2 − U V W = 0. We prove that there exists such that x 2 + y 2 + z 2 = 1, for which U = x y, V = y z, W = z x, with one exception: In case 3.b. Below, we show. 1. In the case where none of U, V, W is 0, we can set x = W U V, y = U V W, z = V W U, it is easy to use to confirm that holds for x, y, z defined this way. 2. Suppose that W is 0. From this implies U 2 V 2
In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids. Other surfaces arise as graphs of functions of two variables. However, surfaces can be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis; the various mathematical notions of surface can be used to model surfaces in the physical world. In mathematics, a surface is a geometrical shape; the most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R3, such as spheres. The exact definition of a surface may depend on the context. In algebraic geometry, a surface may cross itself, while, in topology and differential geometry, it may not.
A surface is a two-dimensional space. In other words, around every point, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles a two-dimensional sphere, latitude and longitude provide two-dimensional coordinates on it; the concept of surface is used in physics, computer graphics, many other disciplines in representing the surfaces of physical objects. For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface. A surface is a topological 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. It is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane; these coordinates are known as local 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 nonempty, second countable, Hausdorff. It is often assumed that the surfaces under consideration are connected; the rest of this article will assume, unless specified otherwise, that a surface is nonempty, second countable, connected. More a surface with boundary is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closure of the upper half-plane H2 in C; these homeomorphisms are 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 such points is known as the boundary of the surface, a one-manifold, that is, the union of closed curves. 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, always non-empty. The closed disk is a simple example of a surface with boundary.
The boundary of the disc is a circle. The term surface used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense. A surface with empty boundary, compact is known as a'closed' surface; the two-dimensional sphere, the two-dimensional torus, the real projective plane are examples of closed surfaces. The Möbius strip is a surface on which the distinction between clockwise and counterclockwise can be defined locally, but not globally. In general, a surface is said to be orientable if it does not contain a homeomorphic copy of the Möbius strip. For example, the sphere and torus are orientable. In differential and algebraic geometry, extra structure is added upon the topology of the surface; this added structures can be a smoothness structure, a Riemannian metric, a complex structure, or an algebraic structure. Surfaces were defined as subspaces of Euclidean spaces; these surfaces were the locus of zeros of certain functions polynomial functions.
Such a definition considered the surface as part of a larger space, as such was termed extrinsic. In the previous section, a surface is defined as a topological space with certain properties, namely Hausdorff and locally Euclidean; this topological space is not considered a subspace of another space. In this sense, the definition given above, the definition that mathematicians use at present, is intrinsic. A surface defined as intrinsic is not required to satisfy the added constraint of being a subspace of Euclidean space, it may seem possible for some surfaces defined intrinsically to not be surfaces in the
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics. By extension, use of complex analysis has applications in engineering fields such as nuclear, aerospace and electrical engineering; as a differentiable function of a complex variable is equal to the sum of its Taylor series, complex analysis is concerned with analytic functions of a complex variable. Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler, Riemann, Cauchy and many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is used throughout analytic number theory. In modern times, it has become popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions.
Another important application of complex analysis is in string theory which studies conformal invariants in quantum field theory. A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are supposed to have a domain that contains a nonempty open subset of the complex plane. For any complex function, the values z from the domain and their images f in the range may be separated into real and imaginary parts: z = x + i y and f = f = u + i v, where x, y, u, v are all real-valued. In other words, a complex function f: C → C may be decomposed into u: R 2 → R and v: R 2 → R, i.e. into two real-valued functions of two real variables. Any complex-valued function f on an arbitrary set X can be considered as an ordered pair of two real-valued functions: or, alternatively, as a vector-valued function from X into R 2; some properties of complex-valued functions are nothing more than the corresponding properties of vector valued functions of two real variables.
Other concepts of complex analysis, such as differentiability are direct generalizations of the similar concepts for real functions, but may have different properties. In particular, every differentiable complex function is analytic, two differentiable functions that are equal in a neighborhood of a point are equal on the intersection of their domain; the latter property is the basis of the principle of analytic continuation which allows extending every real analytic function in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed. Many basic and special complex functions are defined in this way, including exponential functions, logarithmic functions, trigonometric functions. Complex functions that are differentiable at every point of an open subset Ω of the complex plane are said to be holomorphic on Ω. In the context of complex analysis, the derivative of f at z 0 is defined to be f ′ = lim z → z 0 f − f z − z 0, z ∈ C.
Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in different ways compared to their real counterparts. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we