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In 3 dimensions the curl of a vector field is a vector field as is familiar (in 1 and 0 dimensions the curl of a vector field is 0, because there are no non-trivial 2-vectors), while in 4 dimensions the curl of a vector field is, geometrically, at each point an element of the 6-dimensional Lie algebra ().
For a tensor field of order k > 1, the tensor field of order k is defined by the recursive relation = where is an arbitrary constant vector. A tensor field of order greater than one may be decomposed into a sum of outer products, and then the following identity may be used: = ().
The line integral of a vector field over a loop is equal to the surface integral of its curl over the enclosed surface. Stokes' theorem is a special case of the generalized Stokes theorem. [5] [6] In particular, a vector field on can be considered as a 1-form in which case its curl is its exterior derivative, a 2-form.
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a ...
The classical Stokes' theorem relates the surface integral of the curl of a vector field over a surface in Euclidean three-space to the line integral of the vector field over its boundary. It is a special case of the general Stokes theorem (with n = 2 {\displaystyle n=2} ) once we identify a vector field with a 1-form using the metric on ...
This is because of the way the fields are expressed as gradients and curls of the scalar and vector potentials. The homogeneous equations in terms of these potentials involve the divergence of the curl and the curl of the gradient , which are always zero. The other two of Maxwell's equations (the inhomogeneous equations) are the ones that ...
where ⋆ is the Hodge star operator, ♭ and ♯ are the musical isomorphisms, f is a scalar field and F is a vector field. Note that the expression for curl requires ♯ to act on ⋆d(F ♭), which is a form of degree n − 2. A natural generalization of ♯ to k-forms of arbitrary degree allows this expression to make sense for any n.