The following are important identities involving derivatives and integrals in vector calculus. Operator notation Gradient For a function 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 , also called a scalar field, the gradient is the vector field: where are orthogonal unit vectors in arbitrary directions. As the name implies, the gradient is proportional to and points in the direction of the function's most rapid (positive) change. For a vector field 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 of any order k, the gradient is a tensor field of order k + 1. Divergence In Cartesian coordinates, the divergence of a continuously differentiable vector field is the scalar-valued function: As the name implies the divergence is a measure of how much vectors are diverging. The divergence of a tensor field of non-zero order k is written as , 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 is the directional derivative in the direction of multiplied by its magnitude. Specifically, for the outer product of two vectors, Curl In Cartesian coordinates, for the curl is the vector field: where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. As the name implies the curl is a measure of how much nearby vectors tend in a circular direction. In Einstein notation, the vector field has curl given by: where = ±1 or 0 is the Levi-Civita parity symbol. Laplacian In Cartesian coordinates, the Laplacian of a function is The Laplacian is a measure of how much a function is changing over a small sphere centered at the point. For a tensor field, , the Laplacian is generally written as: and is a tensor field of the same order. When the Laplacian is equal to 0, the function is called a harmonic function. That is, Special notations In Feynman subscript notation, where the notation ∇B means the subscripted