Vector calculus identities

The following identities are important in vector calculus: ∇ ψ = ( ∂ ψ ∂ x 1 , … , ∂ ψ ∂ x n ) . {displaystyle abla psi =({ frac {partial psi }{partial x_{1}}},ldots ,{ frac {partial psi }{partial x_{n}}}).} ∇ A = J A = ( ∂ A i ∂ x j ) i j . {displaystyle abla !mathbf {A} =mathbf {J} _{mathbf {A} }=left({frac {partial A_{i}}{partial x_{j}}} ight)_{!ij}.} The following identities are important in vector calculus: For a function f ( x , y , z ) {displaystyle f(x,y,z)} in three-dimensional Cartesian coordinate variables, the gradient is the vector field: where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n variables ψ ( x 1 , … , x n ) {displaystyle psi (x_{1},ldots ,x_{n})} , also called a scalar field, the gradient is the vector field: For a vector field A = ( A 1 , … , A n ) {displaystyle mathbf {A} =(A_{1},ldots ,A_{n})} written as a 1 × n row vector, also called a tensor field of order 1, the gradient or covariant derivative is the n × n Jacobian matrix: For a tensor field A {displaystyle mathbf {A} } of any order k, the gradient grad ⁡ ( A ) = ∇ A {displaystyle operatorname {grad} (mathbf {A} )= abla !mathbf {A} } is a tensor field of order k + 1. In Cartesian coordinates, the divergence of a continuously differentiable vector field F = F x i + F y j + F z k {displaystyle mathbf {F} =F_{x}mathbf {i} +F_{y}mathbf {j} +F_{z}mathbf {k} } is the scalar-valued function: The divergence of a tensor field A {displaystyle mathbf {A} } of non-zero order k, is written as div ⁡ ( A ) = ∇ ⋅ A {displaystyle operatorname {div} (mathbf {A} )= abla cdot mathbf {A} } , a contraction to a tensor field of order k - 1. Specifically, the divergence of a vector is a scalar. The divergence of a higher order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity, where B ⋅ ∇ {displaystyle mathbf {B} cdot abla } is the directional derivative in the direction of B {displaystyle mathbf {B} } multiplied by its magnitude. Specifically, for the outer product of two vectors, In Cartesian coordinates, for F = F x i + F y j + F z k {displaystyle mathbf {F} =F_{x}mathbf {i} +F_{y}mathbf {j} +F_{z}mathbf {k} } the curl is the vector field: