Editing Vector calculus (section)

== Generalizations ==
{{Unreferenced section|date=August 2019}}
Vector calculus can also be generalized to other [[3-manifolds]] and [[higher dimension|higher-dimensional]] spaces.

=== Different 3-manifolds ===
Vector calculus is initially defined for [[Euclidean space|Euclidean 3-space]], <math>\mathbb{R}^3,</math> which has additional structure beyond simply being a 3-dimensional real vector space, namely: a [[norm (mathematics)|norm]] (giving a notion of length) defined via an [[inner product]] (the [[dot product]]), which in turn gives a notion of angle, and an [[orientability|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.

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 ''{{slink|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 [[nondegenerate form]]) and an orientation; this is less data than an isomorphism to Euclidean space, as it does not require a set of coordinates (a frame of reference), which reflects the fact that vector calculus is invariant under rotations (the [[special orthogonal group]] {{math|SO(3)}}).

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 {{math|''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 {{math|0}}, {{math|1}}, {{math|''n'' &minus; 1}} or {{math|''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<ref>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}}</ref> (and, trivially, in dimension 0 or 1) is the curl of a vector field a vector field, and only in 3 or [[seven-dimensional cross product|7]] dimensions can a cross product be defined (generalizations in other dimensionalities either require <math>n-1</math> vectors to yield 1 vector, or are alternative [[Lie algebra]]s, which are more general antisymmetric bilinear products). The generalization of grad and div, and how curl may be generalized is elaborated at ''[[Curl (mathematics)#Generalizations|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 <math>\textstyle{\binom{n}{2}=\frac{1}{2}n(n-1)}</math> dimensions of rotations in {{math|''n''}} dimensions).

There are two important alternative generalizations of vector calculus. The first, [[geometric algebra]], uses [[multivector|{{math|''k''}}-vector]] fields instead of vector fields (in 3 or fewer dimensions, every {{math|''k''}}-vector field can be identified with a scalar function or vector field, but this is not true in higher dimensions). This replaces the cross product, which is specific to 3 dimensions, taking in two vector fields and giving as output a vector field, with the [[exterior product]], which exists in all dimensions and takes in two vector fields, giving as output a bivector (2-vector) field. This product yields [[Clifford algebra]]s as the algebraic structure on vector spaces (with an orientation and nondegenerate form). Geometric algebra is mostly used in generalizations of physics and other applied fields to higher dimensions.

The second generalization uses [[differential form]]s ({{math|''k''}}-covector fields) instead of vector fields or {{math|''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 {{math|''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 {{math|''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.