List of multivariable calculus topics

This is a list of multivariable calculus topics. See also multivariable calculus

Multivariable calculus

Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one....

, vector calculus, list of real analysis topics, list of calculus topics.

Closed and exact differential forms
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 , and an exact form is a differential form that is the exterior derivative of another differential form β...
Contact (mathematics)
Contact (mathematics)
In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. The equivalence classes are generally called jets...
Contour integral
Contour line
Contour line
A contour line of a function of two variables is a curve along which the function has a constant value. In cartography, a contour line joins points of equal elevation above a given level, such as mean sea level...
Critical point (mathematics)
Critical point (mathematics)
In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
Curl (mathematics)
Current (mathematics)
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...
Curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
Curvilinear coordinates
Curvilinear coordinates
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 at each point. This means that one can convert a point given...
D'Alembertian operator
Del
Del
In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...
Differential form
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 forms provide a better definition for integrands in calculus...
Differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
Directional derivative
Directional derivative
In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...
Divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
Divergence theorem
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
Double integral
Equipotential surface
Equipotential surface
Equipotential surfaces are surfaces of constant scalar potential. They are used to visualize an -dimensional scalar potential function in dimensional space...
Euler's theorem on homogeneous functions
Exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
Flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
Frenet-Serret formulas
Frenet-Serret formulas
In vector calculus, the Frenet–Serret formulas describe the kinematic properties of a particle which moves along a continuous, differentiable curve in three-dimensional Euclidean space R3...
Gauss's law
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. Gauss's law states that:...
Gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
Green's theorem
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...
Green's identities
Green's identities
In mathematics, Green's identities are a set of three identities in vector calculus. They are named after the mathematician George Green, who discovered Green's theorem.-Green's first identity:...
Harmonic function
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
Helmholtz decomposition
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 vector field and a...
Hessian matrix
Hessian matrix
In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
Hodge star operator
Inverse function theorem
Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...
Irrotational vector field
Irrotational vector field
In vector calculus a conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential. Conservative vector fields have the property that the line integral from one point to another is independent of the choice of path connecting the two...
Isoperimetry
Isoperimetry
The isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. Isoperimetric literally means "having the same perimeter"...
Jacobian matrix
Lagrange multiplier
Lamellar vector field
Laplacian
Laplacian vector field
Level set
Level set
In mathematics, a level set of a real-valued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....
Line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
Mixed derivatives
Monkey saddle
Monkey saddle
In mathematics, the monkey saddle is the surface defined by the equation z = x^3 - 3xy^2. \, 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
Multiple integral
The multiple integral is a type of definite integral extended to functions of more than one real variable, for example, ƒ or ƒ...
Newtonian potential
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. As such, it is a fundamental object of study in potential theory...
Parametric equation
Parametric equation
In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
Parametric surface
Parametric surface
A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters. Parametric representation is the most general way to specify a surface. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence...
Partial derivative
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...
Partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
Potential
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
Saddle point
Saddle point
In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
Solenoidal vector field
Stokes' theorem
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
Submersion
Submersion
Submersion may refer to:*Being underwater or going underwater: see scuba diving or submarine or :wikt:submerge.*Submersion , in the mathematical sense.*Submersion , an episode of the television series Stargate Atlantis....
Surface integral
Surface integral
In mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...
Symmetry of second derivatives
Symmetry of second derivatives
In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a functionfof n variables...
Taylors theorem
Total derivative
Total derivative
In the mathematical field of differential calculus, the term total derivative has a number of closely related meanings.The total derivative of a function f, of several variables, e.g., t, x, y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative...
Vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
Vector operator
Vector potential
Vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....

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