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# List of multivariable calculus topics

This is a list of multivariable calculus topics. [1]

63 relations: Closed and exact differential forms, Complex lamellar vector field, Conservative vector field, Contact (mathematics), Contour integration, Contour line, Critical point (mathematics), Curl (mathematics), Current (mathematics), Curvature, Curvilinear coordinates, Del, Differential form, Differential operator, Directional derivative, Divergence, Divergence theorem, Equipotential, Exterior derivative, Flux, Frenet–Serret formulas, Gauss's law, Gradient, Green's identities, Green's theorem, Harmonic function, Helmholtz decomposition, Hessian matrix, Hodge star operator, Homogeneous function, Inverse function theorem, Isoperimetric inequality, Jacobian matrix and determinant, Lagrange multiplier, Laplace operator, Laplacian vector field, Level set, Line integral, List of calculus topics, List of real analysis topics, Matrix calculus, Monkey saddle, Multiple integral, Multivariable calculus, Newtonian potential, Parametric equation, Parametric surface, Partial derivative, Partial differential equation, Potential, ... Expand index (13 more) »

## Closed and exact differential forms

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα.

## Complex lamellar vector field

In vector calculus, a complex lamellar vector field is a vector field in three dimensions which is orthogonal to its own curl.

## Conservative vector field

In vector calculus, a conservative vector field is a vector field that is the gradient of some function, known in this context as a scalar potential.

## Contact (mathematics)

In mathematics, two functions have a contact of order k if, at a point P, they have the same value and k equal derivatives.

## Contour integration

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

## Contour line

A contour line (also isocline, isopleth, isarithm, or equipotential curve) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value.

## Critical point (mathematics)

In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0.

## Curl (mathematics)

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space.

## Current (mathematics)

In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Formally currents behave like Schwartz distributions on a space of differential forms.

## Curvature

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.

## Curvilinear coordinates

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved.

## Del

Del, or nabla, is an operator used in mathematics, in particular in vector calculus, as a vector differential operator, usually represented by the nabla symbol &nabla;.

## Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.

## Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

## Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.

## Divergence

In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point.

## Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.

## Equipotential

Equipotential or isopotential in mathematics and physics refers to a region in space where every point in it is at the same potential.

## Exterior derivative

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.

## Flux

Flux describes the quantity which passes through a surface or substance.

## Frenet–Serret formulas

In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a continuous, differentiable curve in three-dimensional Euclidean space ℝ3, or the geometric properties of the curve itself irrespective of any motion.

## Gauss's law

In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field.

In mathematics, the gradient is a multi-variable generalization of the derivative.

## Green's identities

In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act.

## Green's theorem

In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is named after George Green, though its first proof is due to Bernhard Riemann and is the two-dimensional special case of the more general Kelvin–Stokes theorem.

## Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R where U is an open subset of Rn that satisfies Laplace's equation, i.e. everywhere on U. This is usually written as or.

## Helmholtz decomposition

In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition or Helmholtz representation.

## Hessian matrix

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field.

## Hodge star operator

In mathematics, the Hodge isomorphism or Hodge star operator is an important linear map introduced in general by W. V. D. Hodge.

## Homogeneous function

In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

## Inverse function theorem

In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point.

## Isoperimetric inequality

In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area of a set and its volume.

## Jacobian matrix and determinant

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

## Lagrange multiplier

In mathematical optimization, the method of Lagrange multipliers (named after Joseph-Louis Lagrange) is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables).

## Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.

## Laplacian vector field

In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible.

## Level set

In mathematics, a level set of a real-valued function ''f'' of ''n'' real variables is a set of the form that is, a set where the function takes on a given constant value c. When the number of variables is two, a level set is generically a curve, called a level curve, contour line, or isoline.

## Line integral

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.

## List of calculus topics

This is a list of calculus topics.

## List of real analysis topics

This is a list of articles that are considered real analysis topics.

## Matrix calculus

In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.

In mathematics, the monkey saddle is the surface defined by the equation It belongs to the class of saddle surfaces and its name derives from the observation that a saddle for a monkey requires three depressions: two for the legs, and one for the tail.

## Multiple integral

The multiple integral is a definite integral of a function of more than one real variable, for example, or.

## Multivariable calculus

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables, rather than just one.

## Newtonian potential

In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity.

## Parametric equation

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.

## Parametric surface

A parametric surface is a surface in the Euclidean space \Bbb R^3 which is defined by a parametric equation with two parameters \vec r: \Bbb^2 \rightarrow \Bbb^3.

## Partial derivative

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 (as opposed to the total derivative, in which all variables are allowed to vary).

## Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

## Potential

Potential generally refers to a currently unrealized ability.

## Real coordinate space

In mathematics, real coordinate space of dimensions, written R (also written with blackboard bold) is a coordinate space that allows several (''n'') real variables to be treated as a single variable.

In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) of orthogonal function components defining the surface become zero (a stationary point) but are not a local extremum on both axes.

## Solenoidal vector field

In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a '''transverse vector field''') is a vector field v with divergence zero at all points in the field.

## Stokes' theorem

In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes theorem or the Stokes–Cartan theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.

## Submersion (mathematics)

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective.

## Surface integral

In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces.

## Symmetry of second derivatives

In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility under certain conditions (see below) of interchanging the order of taking partial derivatives of a function of n variables.

## Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial.

## Total derivative

In the mathematical field of differential calculus, a total derivative or full derivative of a function f of several variables, e.g., t, x, y, etc., with respect to an exogenous argument, e.g., t, is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function.

## Vector calculus

Vector calculus, or vector analysis, is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3.

## Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.

## Vector operator

A vector operator is a differential operator used in vector calculus.

## Vector potential

In vector calculus, a vector potential is a vector field whose curl is a given vector field.

## References

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