* picture info Divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value. Physical interpretation of divergence In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source at a given point. It is a local measure of its "outgoingness" – the extent t ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Divergence (captions) In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value. Physical interpretation of divergence In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source at a given point. It is a local measure of its "outgoingness" – the extent to ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Vector Calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, ''Vector Analysis''. In the conventional form using cross products, vector calculus does not generalize to higher dimensions ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Del In Cylindrical And Spherical Coordinates This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. Notes * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of ''θ'' and ''φ''): ** The polar angle is denoted by \theta \in , \pi/math>: it is the angle between the ''z''-axis and the radial vector connecting the origin to the point in question. ** The azimuthal angle is denoted by \varphi \in , 2\pi/math>: it is the angle between the ''x''-axis and the projection of the radial vector onto the ''xy''-plane. * The function can be used instead of the mathematical function owing to its domain and image. The classical arctan function has an image of , whereas atan2 is defined to have an image of . Coordinate conversions CAUTION: the operation \arctan\left(\frac\right) must be interpreted as the two-argument inverse tangent, atan2. Unit vector conversions Del for ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Curvilinear Coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name ''curvilinear coordinates'', coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R3) are cylindrical and spherical coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example ''z'' = 0 defines the ''x''-''y'' plane. In the same space, the coordinate surface ''r'' = 1 in spherical coordinates is the surface of a unit sphere, which is curved. The formalism of curviline ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Surface Integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field (that is, a function of position which returns a scalar as a value) over the surface, or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a ''surface'' as shown in the illustration. Surface integrals have applications in physics, particularly with the theories of classical electromagnetism. Surface integrals of scalar fields Assume that ''f'' is a scalar, vector, or tensor field defined on a surface ''S''. To find an explicit formula for the surface integral of ''f'' over ''S'', we need to parameterize ''S'' by defining a system of curvilinear coordinates on ''S'', like the latitude and longitude on a sphere. Let such a parameterizat ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Metric Tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on consists of a metric tensor at each point of that varies smoothly with . A metric tensor is ''positive-definite'' if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined as ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Volume Element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV = \rho(u_1,u_2,u_3)\,du_1\,du_2\,du_3 where the u_i are the coordinates, so that the volume of any set B can be computed by :\operatorname(B) = \int_B \rho(u_1,u_2,u_3)\,du_1\,du_2\,du_3. For example, in spherical coordinates dV = u_1^2\sin u_2\,du_1\,du_2\,du_3, and so \rho = u_1^2\sin u_2. The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the area element, and in this setting it is useful for doing surface integrals. Under changes of coordinates, the volume element changes by the absolute value of the Jacobian determinant of the coordinate transformation (by the change of variables formula). This fact allows volume elements to be defined as a kind of measure on a manifold. On an ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info YouTube YouTube is a global online video sharing and social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by Google, and is the second most visited website, after Google Search. YouTube has more than 2.5 billion monthly users who collectively watch more than one billion hours of videos each day. , videos were being uploaded at a rate of more than 500 hours of content per minute. In October 2006, YouTube was bought by Google for \$1.65 billion. Google's ownership of YouTube expanded the site's business model, expanding from generating revenue from advertisements alone, to offering paid content such as movies and exclusive content produced by YouTube. It also offers YouTube Premium, a paid subscription option for watching content without ads. YouTube also approved creators to participate in Google's AdSense program, which seeks to generate more revenue for both parties. YouTube ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines such as number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl contributed to an exceptionally wide range of mathematical fields, including works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. Freeman Dyson wrote that Weyl alone bore co ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Einstein Notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916. Introduction Statement of convention According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see Free and bound variables), it implies summation of that term over all the values of the index. So where the indices can range over the set , : y = \sum_^3 c_i x^i = c_1 x^1 + c_2 x^2 + c_3 x^3 is simplified by the convention to: : y = c_i x^i The upper indices are not exponents but are indices ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Tensor Field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a pure number plus a direction, like velocity), a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor is defined on a vector fields set over a module , we call a tensor field on . Many mathematical structures called "tensors" are also tensor fields. For example, the Riemann curvature tensor is a tensor ''field'' as it is defined on a manifold: it is named after Bernhard Riemann, and associates a tensor to each point of a Riemannian m ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Spherical Coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measured from a fixed zenith direction, and the ''azimuthal angle'' of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system. The radial distance is also called the ''radius'' or ''radial coordinate''. The polar angle may be called ''colatitude'', ''zenith angle'', '' normal angle'', or '' inclination angle''. When radius is fixed, the two angular coordinates make a coordinate system on the sphere sometimes called spherical polar coordinates. The use of symbols and the order of the coordinates differs among sources and disciplines. This article will ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]