In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More each point of an n-dimensional manifold has a neighbourhood, homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold. One-dimensional manifolds include circles, but not figure eights. Two-dimensional manifolds are called surfaces. Examples include the plane, the sphere, the torus, which can all be embedded in three dimensional real space, but the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space. Although a manifold locally resembles Euclidean space, meaning that every point has a neighbourhood homeomorphic to an open subset of Euclidean space, globally it may not: manifolds in general are not homeomorphic to Euclidean space. For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane.
When a region appears in two neighbouring charts, the two representations do not coincide and a transformation is needed to pass from one to the other, called a transition map. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space. Manifolds arise as solution sets of systems of equations and as graphs of functions. Manifolds can be equipped with additional structure. One important class of manifolds is the class of differentiable manifolds. A Riemannian metric on a manifold allows angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. A surface is a two dimensional manifold, meaning that it locally resembles the Euclidean plane near each point. For example, the surface of a globe can be described by a collection of maps, which together form an atlas of the globe.
Although no individual map is sufficient to cover the entire surface of the globe, any place in the globe will be in at least one of the charts. Many places will appear in more than one chart. For example, a map of North America will include parts of South America and the Arctic circle; these regions of the globe will be described in full in separate charts, which in turn will contain parts of North America. There is a relation between adjacent charts, called a transition map that allows them to be patched together to cover the whole of the globe. Describing the coordinate charts on surfaces explicitly requires knowledge of functions of two variables, because these patching functions must map a region in the plane to another region of the plane. However, one-dimensional examples of manifolds can be described with functions of a single variable only. Manifolds have applications in computer-graphics and augmented-reality given the need to associate pictures to coordinates. In an augmented reality setting, a picture can be seen as something associated with a coordinate and by using sensors for detecting movements and rotation one can have knowledge of how the picture is oriented and placed in space.
After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Consider, for instance, the top part of the unit circle, x2 + y2 = 1, where the y-coordinate is positive. Any point of this arc can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous, invertible, mapping from the upper arc to the open interval: χ t o p = x; such functions along with the open regions they map are called charts. There are charts for the bottom and right parts of the circle: χ b o t t o m = x χ l e f t = y χ r i g h t = y. Together, these parts cover the four charts form an atlas for the circle; the top and right charts, χ t o
Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word refers to visible light, the visible spectrum, visible to the human eye and is responsible for the sense of sight. Visible light is defined as having wavelengths in the range of 400–700 nanometres, or 4.00 × 10−7 to 7.00 × 10−7 m, between the infrared and the ultraviolet. This wavelength means a frequency range of 430–750 terahertz; the main source of light on Earth is the Sun. Sunlight provides the energy that green plants use to create sugars in the form of starches, which release energy into the living things that digest them; this process of photosynthesis provides all the energy used by living things. Another important source of light for humans has been fire, from ancient campfires to modern kerosene lamps. With the development of electric lights and power systems, electric lighting has replaced firelight; some species of animals generate their own light, a process called bioluminescence.
For example, fireflies use light to locate mates, vampire squids use it to hide themselves from prey. The primary properties of visible light are intensity, propagation direction, frequency or wavelength spectrum, polarization, while its speed in a vacuum, 299,792,458 metres per second, is one of the fundamental constants of nature. Visible light, as with all types of electromagnetic radiation, is experimentally found to always move at this speed in a vacuum. In physics, the term light sometimes refers to electromagnetic radiation of any wavelength, whether visible or not. In this sense, gamma rays, X-rays and radio waves are light. Like all types of EM radiation, visible light propagates as waves. However, the energy imparted by the waves is absorbed at single locations the way particles are absorbed; the absorbed energy of the EM waves is called a photon, represents the quanta of light. When a wave of light is transformed and absorbed as a photon, the energy of the wave collapses to a single location, this location is where the photon "arrives."
This is. This dual wave-like and particle-like nature of light is known as the wave–particle duality; the study of light, known as optics, is an important research area in modern physics. EM radiation, or EMR, is classified by wavelength into radio waves, infrared, the visible spectrum that we perceive as light, ultraviolet, X-rays, gamma rays; the behavior of EMR depends on its wavelength. Higher frequencies have shorter wavelengths, lower frequencies have longer wavelengths; when EMR interacts with single atoms and molecules, its behavior depends on the amount of energy per quantum it carries. EMR in the visible light region consists of quanta that are at the lower end of the energies that are capable of causing electronic excitation within molecules, which leads to changes in the bonding or chemistry of the molecule. At the lower end of the visible light spectrum, EMR becomes invisible to humans because its photons no longer have enough individual energy to cause a lasting molecular change in the visual molecule retinal in the human retina, which change triggers the sensation of vision.
There exist animals that are sensitive to various types of infrared, but not by means of quantum-absorption. Infrared sensing in snakes depends on a kind of natural thermal imaging, in which tiny packets of cellular water are raised in temperature by the infrared radiation. EMR in this range causes molecular vibration and heating effects, how these animals detect it. Above the range of visible light, ultraviolet light becomes invisible to humans because it is absorbed by the cornea below 360 nm and the internal lens below 400 nm. Furthermore, the rods and cones located in the retina of the human eye cannot detect the short ultraviolet wavelengths and are in fact damaged by ultraviolet. Many animals with eyes that do not require lenses are able to detect ultraviolet, by quantum photon-absorption mechanisms, in much the same chemical way that humans detect visible light. Various sources define visible light as narrowly as 420–680 nm to as broadly as 380–800 nm. Under ideal laboratory conditions, people can see infrared up to at least 1050 nm.
Plant growth is affected by the color spectrum of light, a process known as photomorphogenesis. The speed of light in a vacuum is defined to be 299,792,458 m/s; the fixed value of the speed of light in SI units results from the fact that the metre is now defined in terms of the speed of light. All forms of electromagnetic radiation move at this same speed in vacuum. Different physicists have attempted to measure the speed of light throughout history. Galileo attempted to measure the speed of light in the seventeenth century. An early experiment to measure the speed of light was conducted by Ole Rømer, a Danish physicist, in 1676. Using a telescope, Rømer observed one of its moons, Io. Noting discrepancies in the apparent period of Io's orbit, he calculated that light takes about 22 minutes to traverse the diameter of Earth's orbit. However, its size was not known at that time. If Rømer had known the diameter of the Earth's orbit, he would have calculated a speed of 227,000,000 m/s. Another, more accurate, measurement of the speed of light was performed in Europe by Hippolyte Fizeau in 1849.
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
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp
In physics, a force is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass i.e. to accelerate. Force can be described intuitively as a push or a pull. A force has both direction, making it a vector quantity, it is measured in the SI unit of newtons and represented by the symbol F. The original form of Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. If the mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, is inversely proportional to the mass of the object. Concepts related to force include: thrust. In an extended body, each part applies forces on the adjacent parts; such internal mechanical stresses cause no acceleration of that body as the forces balance one another. Pressure, the distribution of many small forces applied over an area of a body, is a simple type of stress that if unbalanced can cause the body to accelerate.
Stress causes deformation of solid materials, or flow in fluids. Philosophers in antiquity used the concept of force in the study of stationary and moving objects and simple machines, but thinkers such as Aristotle and Archimedes retained fundamental errors in understanding force. In part this was due to an incomplete understanding of the sometimes non-obvious force of friction, a inadequate view of the nature of natural motion. A fundamental error was the belief that a force is required to maintain motion at a constant velocity. Most of the previous misunderstandings about motion and force were corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Sir Isaac Newton formulated laws of motion that were not improved for nearly three hundred years. By the early 20th century, Einstein developed a theory of relativity that predicted the action of forces on objects with increasing momenta near the speed of light, provided insight into the forces produced by gravitation and inertia.
With modern insights into quantum mechanics and technology that can accelerate particles close to the speed of light, particle physics has devised a Standard Model to describe forces between particles smaller than atoms. The Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are: strong, electromagnetic and gravitational. High-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction. Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines; the mechanical advantage given by a simple machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces culminated in the work of Archimedes, famous for formulating a treatment of buoyant forces inherent in fluids.
Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the terrestrial sphere contained four elements that come to rest at different "natural places" therein. Aristotle believed that motionless objects on Earth, those composed of the elements earth and water, to be in their natural place on the ground and that they will stay that way if left alone, he distinguished between the innate tendency of objects to find their "natural place", which led to "natural motion", unnatural or forced motion, which required continued application of a force. This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles, such as the flight of arrows; the place where the archer moves the projectile was at the start of the flight, while the projectile sailed through the air, no discernible efficient cause acts on it.
Aristotle was aware of this problem and proposed that the air displaced through the projectile's path carries the projectile to its target. This explanation demands a continuum like air for change of place in general. Aristotelian physics began facing criticism in medieval science, first by John Philoponus in the 6th century; the shortcomings of Aristotelian physics would not be corrected until the 17th century work of Galileo Galilei, influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion, he showed that the bodies were accelerated by gravity to an extent, independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction. Sir Isaac Newton described the motion of all objects using the concepts of inertia and force, in doing so he found they obey certain conservation laws.
In 1687, Newton published his thesis Philosophiæ Naturalis Principia Mathematica. In this work Newton set out three laws of motion that to this day are t