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In mathematics, there are many possible generalizations of the derivative, i.e. the fundamental construction of the differential calculus. Mathematics is the study of quantity, structure, space and change. ...
In mathematics, the derivative is one of the two central concepts of calculus. ...
In mathematics, the derivative of a function is one of the two central concepts of calculus. ...
Multivariable calculus
The derivative is often met for the first time as an operation on a single real function of a single real variable. One of the simplest settings for generalizations is to vector valued functions of several variables (most often the domain forms a vector space as well). This is the field of multivariable calculus. Multivariate calculus is a means of analyzing deterministic systems with multiple degrees of freedom. ...
The partial derivative is defined almost identically with the derivative of a single real function, except it recognizes the choice of independent variable being made. In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. ...
The total derivative of a function is used to indicate that a function may have both explicit and implicit dependence on a variable. The total derivative should take both possible sources of variation into account, whereas a partial derivative should only see explicit dependence. In mathematics, a total derivative may mean either (i) a differential operator involving the sum of all the partial derivatives with respect to all variables in a problem, or be used compatibly (ii) to express the exterior derivative d, as applied to differential forms, and in particular as applied to...
The convective derivative takes into account changes due to time dependence and motion through space along vector field. The convective derivative, also known as the Lagrangian derivative, is a derivative taken with a respect to a coordinate system moving with velocity u, and is often used in fluid mechanics. ...
For real valued functions from Rn to R, the gradient yields a vector each component of which is a partial derivative. This can be used to calculate directional derivatives of scalar functions or normal directions. In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change. ...
In mathematics, the directional derivative of a multivariate differentiable function along a given unit vector intuitively represents the rate of change of the function in the direction of that vector. ...
Scalar is a concept that has meaning in mathematics, physics, and computing. ...
For vector valued functions Rn to Rn, the divergence gives a measure of how much "source" or "sink" near a point there is. It can be used to calculate flux by Stokes theorem. In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...
The flux visualized. ...
The Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...
For vector valued functions R3 to R3, the curl measures how much "rotation" a vector field has near a point. cURL is a command line tool for transferring files with URL syntax, supporting FTP, FTPS, HTTP, HTTPS, Gopher, Telnet, DICT, FILE and LDAP. cURL supports HTTPS certificates, HTTP POST, HTTP PUT, FTP uploading, Kerberos, HTTP form based upload, proxies, cookies, user+password authentication, file transfer resume, http proxy tunneling and...
Rotation is the movement of a body in such a way that the distance between a certain fixed point and any given point of that body remains constant. ...
For any function from Rn to Rm, the Jacobian is a matrix containing all the partial derivatives of all the components of the function. Its determinant is used in the change of variables formula of integration. In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...
In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ...
Higher order derivatives Another simple generalization one can make to the derivative is to simply apply it more than once, obtaining second order derivative (and higher), as defined in the article on derivatives. This notion can be generalized.
In addition to n-th derivatives for any natural number n, using various methods, one can take derivatives to fractional or negative powers. The -1 order derivative will then correspond to the integral, whence the term differintegral. The study of different possible definitions and notions of derivatives to nonnatural numbered powers is known as fractional calculus. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
In mathematics, the differintegral is the combined differentiation/integration operator used in fractional calculus. ...
In mathematics, fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers of the differential operator D = d/dx and the integration operator I. In this context powers refer iterative application, in the same sense that f2(x) = f(f(x)). For example...
Higher order derivatives in multivariable calculus There are several different vector and scalar valued second order derivatives encountered in multivariable calculus.
The Laplacian is the divergence of the gradient of a scalar function on Rn. The d'Alembertian is defined similar to the Laplacian, but using the indefinite metric of Minkowski space, instead of the Euclidean dot product of Rn. In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...
In special relativity, electromagnetism and wave theory, the dAlembert operator, also called dAlembertian, is the Laplace operator of Minkowski space. ...
See: International System of Units, colloquially called the Metric System, and also metrication. ...
In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...
In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
In mathematics, the dot product (also known as the scalar product and the inner product) is a sesquilinear function (·) : V × V → F, where V is a vector space over the field F, having some further properties. ...
The Hessian matrix is a matrix of second order partial derivatives of a scalar function, used in calculations in Morse theory. In mathematics, the Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function. ...
In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. ...
Algebra A derivation is a linear map on a ring or algebra which satisfies the Leibniz law (product rule). They are studied in a purely algebraic setting in differential Galois theory, but also turn up in many other areas, where they often agree with less algebraic definitions of derivatives. In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...
In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...
In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...
In mathematics, the product rule of calculus, which is also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...
Motivation and Basic Idea In mathematics, the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. ...
Differential topology In differential topology, a vector field may be defined as a derivation on the ring of smooth functions on a manifold, and a tangent vector may be defined as a derivation at a point. This allows the abstraction of the notion of a directional derivative of a scalar function to general manifolds. For manifolds that are subsets of Rn, this tangent vector will agree with the directional derivative defined above. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...
In mathematics, a smooth function is one that is infinitely differentiable, i. ...
This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ...
In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...
A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...
The pushforward of a map between manifolds is the induced map between tangent spaces of those maps. It abstracts the Jacobian matrix. In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ...
In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...
On the exterior algebra of differential forms over a smooth manifold, the exterior derivative is the unique linear map which satisfies a graded version of the Leibniz law and squares to zero. It is a grade 1 derivation on the exterior algebra. In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...
A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...
In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...
In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...
The Lie derivative is the rate of change of one vector field in the direction of another vector field. It is an example of a Lie bracket (vector fields form the Lie algebra of the diffeomorphism group of the manifold). It is a grade 0 derivation on the algebra. In mathematics, a Lie derivative is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented by vector fields, as...
A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...
In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...
The inner derivative is a grade -1 derivation on the exterior algebra of forms. Together, the exterior derivative, the Lie derivative, and the inner derivative generate a Lie superalgebra. In mathematics, a Lie superalgebra is a kind of generalisation of a Lie algebra. ...
Differential geometry In differential geometry, the covariant derivative makes a choice for taking directional derivatives of vector fields along curves. This extends the directional derivative of scalar functions to sections of vector bundles or principal bundles. In Riemannian geometry, the existence of a metric chooses a unique preferred torsion-free covariant derivative, known as the Levi-Civita connection. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...
In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves...
In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ...
In mathematics, the term torsion has several meanings, mostly unrelated to each other. ...
In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ...
The exterior covariant derivative extends the exterior derivative to vector valued forms. In differential geometry, the connection form describes connection on principal bundles (or vector bundles). ...
Complex analysis In complex analysis, the central objects of study are holomorphic functions, which are complex-valued functions on the complex numbers satisfying a suitably extended definition of differentiability. Complex analysis is the branch of mathematics investigating holomorphic functions, i. ...
Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
The Schwarzian derivative describes how a complex function is approximated by a fractional-linear map, in much the same way that a normal derivative describes how a function is approximated by a linear map. In mathematics, the Schwarzian derivative is a certain operator that is invariant under all linear fractional transformations. ...
Functional analysis In functional analysis, the functional derivative defines the derivative with respect to a function of a functional on a space of functions. This is an extension of the directional derivative to an infinite dimensional vector space. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...
In mathematics and theoretical physics, the functional derivative is a generalization of the usual derivative that arises in the calculus of variations. ...
Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...
The Fréchet derivative allows the extension of the directional derivative to a general Banach space. In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...
In measure theory, the Radon-Nikodym derivative generalizes the Jacobian, used for changing variables, to measures. It expresses one measure μ in terms of another measure ν (under certain conditions). In mathematics, a measure is a function that assigns a number, e. ...
In mathematics, the Radon-Nikodym theorem is a result in functional analysis that states that if a measure Q is absolutely continuous with respect to another sigma-finite measure P then there is a measurable function f, taking values in [0,∞], on the underlying space such that for any measurable...
In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...
The derivative also admits a generalization to the space of distributions on a space of functions using integration by parts against a suitably well-behaved subspace. In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...
Algebraic geometry In algebraic geometry, the Kähler differential allows the definition of the exterior derivative to be extended to arbitrary algebraic varieties, instead of just smooth manifolds. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
In mathematics, the Kähler differentials are a universal construction Ω1S/R associated to a ring homomorphism of commutative rings, φ:R → S, that provides an analogue of the construction of differential forms (1-forms). ...
This article is about algebraic varieties. ...
Quantum groups In the area of quantum groups, the q-derivative is a q-deformation of the normal derivative of a function. In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes . (Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ...
Other generalizations It may be possible to combine two or more of the above different notions of extension or abstraction of the original derivative. For example, in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some of the features of a functional derivative and the covariant derivative. In mathematics, a Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth and satisfies the following property: For each point x of M, and for every vector v in the tangent...
In mathematics, something is said to occur locally in the category of topological spaces if it occurs on small enough open sets. ...
In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...
In mathematics and theoretical physics, the functional derivative is a generalization of the usual derivative that arises in the calculus of variations. ...
In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...
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