Vector calculus can also be generalized to other
3-manifolds and
higher-dimensional spaces.
Different 3-manifolds Vector calculus is initially defined for
Euclidean 3-space, \mathbb{R}^3, 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. 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. 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 ). 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 -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 , , or 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 (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 n-1 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 \textstyle{\binom{n}{2}=\frac{1}{2}n(n-1)} dimensions of rotations in dimensions). There are two important alternative generalizations of vector calculus. The first,
geometric algebra, uses
-vector fields instead of vector fields (in 3 or fewer dimensions, every -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 algebras 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 forms (-covector fields) instead of vector fields or -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 -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 -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 ==