Vector calculus

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Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, ${\displaystyle \mathbb {R} ^{3}.}$ The term vector calculus is sometimes used as a synonym for the broader subject of

Vector calculus was developed from the theory of

A

A

In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields and scalar fields, except that they change sign under an orientation-reversing map: for example, the curl of a vector field is a pseudovector field, and if one reflects a vector field, the curl points in the opposite direction. This distinction is clarified and elaborated in geometric algebra, as described below.

Vector algebra

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

Notations in vector calculus

Operation	Notation	Description
Vector addition	${\displaystyle \mathbf {v} _{1}+\mathbf {v} _{2}}$	Addition of two vectors, yielding a vector.
Scalar multiplication	${\displaystyle a\mathbf {v} }$	Multiplication of a scalar and a vector, yielding a vector.
Dot product	${\displaystyle \mathbf {v} _{1}\cdot \mathbf {v} _{2}}$	Multiplication of two vectors, yielding a scalar.
Cross product	${\displaystyle \mathbf {v} _{1}\times \mathbf {v} _{2}}$	Multiplication of two vectors in ${\displaystyle \mathbb {R} ^{3}}$, yielding a (pseudo)vector.

Also commonly used are the two triple products:

Vector calculus triple products

Operation	Notation	Description
Scalar triple product	${\displaystyle \mathbf {v} _{1}\cdot \left(\mathbf {v} _{2}\times \mathbf {v} _{3}\right)}$	The dot product of the cross product of two vectors.
Vector triple product	${\displaystyle \mathbf {v} _{1}\times \left(\mathbf {v} _{2}\times \mathbf {v} _{3}\right)}$	The cross product of the 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 (${\displaystyle \nabla }$), also known as "nabla". The three basic vector operators are:^[2]

Differential operators in vector calculus

Operation	Notation	Description	Notational analogy	Domain/Range
Gradient	${\displaystyle \operatorname {grad} (f)=\nabla f}$	Measures the rate and direction of change in a scalar field.	Scalar multiplication	Maps scalar fields to vector fields.
Divergence	${\displaystyle \operatorname {div} (\mathbf {F} )=\nabla \cdot \mathbf {F} }$	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	${\displaystyle \operatorname {curl} (\mathbf {F} )=\nabla \times \mathbf {F} }$	Measures the tendency to rotate about a point in a vector field in ${\displaystyle \mathbb {R} ^{3}}$.	Cross product	Maps vector fields to (pseudo)vector fields.
f denotes a scalar field and F denotes a vector field

Also commonly used are the two Laplace operators:

Laplace operators in vector calculus

Operation	Notation	Description	Domain/Range
Laplacian	${\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f}$	Measures the difference between the value of the scalar field with its average on infinitesimal balls.	Maps between scalar fields.
Vector Laplacian	${\displaystyle \nabla ^{2}\mathbf {F} =\nabla (\nabla \cdot \mathbf {F} )-\nabla \times (\nabla \times \mathbf {F} )}$	Measures the difference between the value of the vector field with its average on infinitesimal balls.	Maps between vector fields.
f denotes a scalar field and F 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:

Integral theorems of vector calculus

Theorem	Statement	Description
Gradient theorem	${\displaystyle \int _{L\subset \mathbb {R} ^{n}}\!\!\!\nabla \varphi \cdot d\mathbf {r} \ =\ \varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)\ \ {\text{ for }}\ \ L=L[p\to q]}$	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	${\displaystyle \underbrace {\int \!\cdots \!\int _{V\subset \mathbb {R} ^{n}}} _{n}(\nabla \cdot \mathbf {F} )\,dV\ =\ \underbrace {\oint \!\cdots \!\oint _{\partial V}} _{n-1}\mathbf {F} \cdot d\mathbf {S} }$	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	${\displaystyle \iint _{\Sigma \subset \mathbb {R} ^{3}}(\nabla \times \mathbf {F} )\cdot d\mathbf {\Sigma } \ =\ \oint _{\partial \Sigma }\mathbf {F} \cdot d\mathbf {r} }$	The integral of the curl of a vector field over a surface Σ in ${\displaystyle \mathbb {R} ^{3}}$ is equal to the circulation of the vector field around the closed curve bounding the surface.
${\displaystyle \varphi }$ denotes a scalar field and F denotes a vector field

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

Green's theorem of vector calculus

Theorem	Statement	Description
Green's theorem	${\displaystyle \iint _{A\,\subset \mathbb {R} ^{2}}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)dA\ =\ \oint _{\partial A}\left(L\,dx+M\,dy\right)}$	The integral of the divergence (or curl) of a vector field over some region A in ${\displaystyle \mathbb {R} ^{2}}$ equals the flux (or circulation) of the vector field over the closed curve bounding the region.
For divergence, F = (M, −L). For curl, F = (L, M, 0). L and M are functions of (x, y).

Applications

Linear approximations

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

${\displaystyle f(x,y)\ \approx \ f(a,b)+{\tfrac {\partial f}{\partial x}}(a,b)\,(x-a)+{\tfrac {\partial f}{\partial y}}(a,b)\,(y-b).}$

The right-hand side is the equation of the plane tangent to the graph of z = f(x, y) at (a, b).

Optimization

For a continuously differentiable function of several real variables, a point P (that is, a set of values for the input variables, which is viewed as a point in Rⁿ) is critical if all of the partial derivatives of the function are zero at P, or, equivalently, if its gradient is zero. The critical values are the values of the function at the critical points.

If the function is

of second derivatives.

By

maxima and minima

of a differentiable function occur at critical points. Therefore, to find the local maxima and minima, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.

Generalizations

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Vector calculus can also be generalized to other

higher-dimensional

spaces.

Different 3-manifolds

Vector calculus is initially defined for Euclidean 3-space, ${\displaystyle \mathbb {R} ^{3},}$ which has additional structure beyond simply being a 3-dimensional real vector space, namely: a

The gradient and divergence require only the inner product, while the curl and the cross product also requires the handedness of the coordinate system to be taken into account (see Cross product § Handedness for more detail).

Vector calculus can be defined on other 3-dimensional real vector spaces if they have an inner product (or more generally a symmetric

More generally, vector calculus can be defined on any 3-dimensional oriented Riemannian manifold, or more generally pseudo-Riemannian manifold. This structure simply means that the tangent space at each point has an inner product (more generally, a symmetric nondegenerate form) and an orientation, or more globally that there is a symmetric nondegenerate metric tensor and an orientation, and works because vector calculus is defined in terms of tangent vectors at each point.

Other dimensions

Most of the analytic results are easily understood, in a more general form, using the machinery of differential geometry, of which vector calculus forms a subset. Grad and div generalize immediately to other dimensions, as do the gradient theorem, divergence theorem, and Laplacian (yielding harmonic analysis), while curl and cross product do not generalize as directly.

From a general point of view, the various fields in (3-dimensional) vector calculus are uniformly seen as being k-vector fields: scalar fields are 0-vector fields, vector fields are 1-vector fields, pseudovector fields are 2-vector fields, and pseudoscalar fields are 3-vector fields. In higher dimensions there are additional types of fields (scalar, vector, pseudovector or pseudoscalar corresponding to 0, 1, n − 1 or n dimensions, which is exhaustive in dimension 3), so one cannot only work with (pseudo)scalars and (pseudo)vectors.

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^[3] (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 ${\displaystyle n-1}$ vectors to yield 1 vector, or are alternative

${\displaystyle \textstyle {{\binom {n}{2}}={\frac {1}{2}}n(n-1)}}$

There are two important alternative generalizations of vector calculus. The first,

The second generalization uses differential forms (k-covector fields) instead of vector fields or k-vector fields, and is widely used in mathematics, particularly in differential geometry, geometric topology, and harmonic analysis, in particular yielding Hodge theory on oriented pseudo-Riemannian manifolds. From this point of view, grad, curl, and div correspond to the exterior derivative of 0-forms, 1-forms, and 2-forms, respectively, and the key theorems of vector calculus are all special cases of the general form of Stokes' theorem.

From the point of view of both of these generalizations, vector calculus implicitly identifies mathematically distinct objects, which makes the presentation simpler but the underlying mathematical structure and generalizations less clear. From the point of view of geometric algebra, vector calculus implicitly identifies k-vector fields with vector fields or scalar functions: 0-vectors and 3-vectors with scalars, 1-vectors and 2-vectors with vectors. From the point of view of differential forms, vector calculus implicitly identifies k-forms with scalar fields or vector fields: 0-forms and 3-forms with scalar fields, 1-forms and 2-forms with vector fields. Thus for example the curl naturally takes as input a vector field or 1-form, but naturally has as output a 2-vector field or 2-form (hence pseudovector field), which is then interpreted as a vector field, rather than directly taking a vector field to a vector field; this is reflected in the curl of a vector field in higher dimensions not having as output a vector field.

See also

References

Citations