Vector calculus

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Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes 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.


Basic objects

Scalar fields

A scalar field associates a scalar value to every point in a space. The scalar is a mathematical number representing a physical quantity. Examples of scalar fields in applications include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.

Vector fields

A vector field is an assignment of a vector to each point in a space.[1] A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. This can be used, for example, to calculate work done over a line.

Vectors and pseudovectors


Vector algebra

The algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then globally applied to a vector field. The basic algebraic operations consist of:

Operation Notation Description
Vector addition Addition of two vectors, yielding a vector.
Scalar multiplication Multiplication of a scalar and a vector, yielding a vector.
Dot product Multiplication of two vectors, yielding a scalar.
Cross product Multiplication of two vectors in , yielding a (pseudo)vector.

Also commonly used are the two triple products:

Operation Notation Description
Scalar triple product The dot product of a vector and a cross product of two vectors.
Vector triple product The cross product of a vector and a cross product of two vectors.

Operators and theorems

Differential operators

Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator (), also known as "nabla". The three basic vector operators are:

Operation Notation Description Notational
Gradient Measures the rate and direction of change in a scalar field. Scalar multiplication Maps scalar fields to vector fields.
Divergence Measures the scalar of a source or sink at a given point in a vector field. Dot product Maps vector fields to scalar fields.
Curl Measures the tendency to rotate about a point in a vector field in . Cross product Maps vector fields to (pseudo)vector fields.
denotes a scalar field and denotes a vector field

Also commonly used are the two Laplace operators:

Operation Notation Description Domain/Range
denotes a scalar field and denotes a vector field

A quantity called the Jacobian matrix is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration.

Integral theorems

The three basic vector operators have corresponding theorems which generalize the fundamental theorem of calculus to higher dimensions:

Theorem Statement Description
Gradient theorem The line integral of the gradient of a scalar field over a curve L is equal to the change in the scalar field between the endpoints p and q of the curve.
Divergence theorem The integral of the divergence of a vector field over an n-dimensional solid V is equal to the flux of the vector field through the (n−1)-dimensional closed boundary surface of the solid.
Curl (Kelvin–Stokes) theorem The integral of the curl of a vector field over a surface Σ in is equal to the circulation of the vector field around the closed curve bounding the surface.
denotes a scalar field and denotes a vector field

In two dimensions, the divergence and curl theorems reduce to the Green's theorem:

Theorem Statement Description
Green's theorem The integral of the divergence (or curl) of a vector field over some region A in equals the flux (or circulation) of the vector field over the closed curve bounding the region.
For divergence, . For curl, . L and M are functions of (x, y).


Linear approximations

Linear approximations are used to replace complicated functions with linear functions that are almost the same. Given a differentiable function with real values, one can approximate for close to by the formula






Physics and engineering

Vector calculus is particularly useful in studying:


Different 3-manifolds

Vector calculus is initially defined for Euclidean 3-space, which has additional structure beyond simply being a 3-dimensional real vector space, namely: a norm (giving a notion of length) defined via an inner product (the dot product), which in turn gives a notion of angle, and an orientation, which gives a notion of left-handed and right-handed. These structures give rise to a volume form, and also the cross product, which is used pervasively in vector calculus.




Other dimensions



In any dimension, assuming a nondegenerate form, grad of a scalar function is a vector field, and div of a vector field is a scalar function, but only in dimension 3 or 7[2] (and, trivially, in dimension 0 or 1) is the curl of a vector field a vector field, and only in 3 or 7 dimensions can a cross product be defined (generalizations in other dimensionalities either require vectors to yield 1 vector, or are alternative Lie algebras, which are more general antisymmetric bilinear products). The generalization of grad and div, and how curl may be generalized is elaborated at Curl: Generalizations; in brief, the curl of a vector field is a bivector field, which may be interpreted as the special orthogonal Lie algebra of infinitesimal rotations; however, this cannot be identified with a vector field because the dimensions differ – there are 3 dimensions of rotations in 3 dimensions, but 6 dimensions of rotations in 4 dimensions (and more generally dimensions of rotations in n dimensions).




See also



  1. ^ Galbis, Antonio & Maestre, Manuel (2012). Vector Analysis Versus Vector Calculus. Springer. p. 12. ISBN 978-1-4614-2199-3.CS1 maint: uses authors parameter (link)
  2. ^ Lizhong Peng & Lei Yang (1999) "The curl in seven dimensional space and its applications", Approximation Theory and Its Applications 15(3): 66 to 80 doi:10.1007/BF02837124


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