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Interchanging the vector field v and ∇ operator, we arrive at the cross product of a vector field with curl of a vector field: = () , where ∇ F is the Feynman subscript notation, which considers only the variation due to the vector field F (i.e., in this case, v is treated as being constant in space).
If a vector field has negative divergence in some area, there will be field lines ending at points in that area. The Kelvin–Stokes theorem shows that field lines of a vector field with zero curl (i.e., a conservative vector field, e.g. a gravitational field or an electrostatic field) cannot be closed loops. In other words, curl is always ...
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: = ().
A vector field V defined on an open set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that = = (,,, …,). The associated flow is called the gradient flow , and is used in the method of gradient descent .
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 ...
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, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
Although the magnetic field, , is a pseudovector (also called axial vector), the vector potential, , is a polar vector. [6] This means that if the right-hand rule for cross products were replaced with a left-hand rule, but without changing any other equations or definitions, then B {\displaystyle \mathbf {B} } would switch signs, but A would ...
The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve