In the three-dimensional Cartesian coordinate variables, the gradient of some function f(x,y,z) is given by:
where i, j, k are the standard unit vectors for the x, y, z-axes.
More generally, for a function of n variables ψ(x1,…,xn), also called a scalar field, the gradient is the vector field:
where ei are orthogonal unit vectors in arbitrary directions.
For a vector field A=(A1,…,An) 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:
The gradient of a tensor field, A of any order k, is generally written as: grad(A)=∇A and is a tensor field of order k + 1.
In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field F=Fxi+Fyj+Fzk is the scalar-valued function:
The divergence of a tensor field A of non-zero order k is written as
and is 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⋅∇ is the directional derivative in the direction of B multiplied by its magnitude. Specifically, for the outer product of two vectors,
In Cartesian coordinates, for F=Fxi+Fyj+Fzk the curl is the vector field:
where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively.
In Einstein notation, the vector field F=(F1 F2 F3)has curl given by:
where ε is the Levi-Civita parity symbol.
In Cartesian coordinates, the Laplacian of a function f(x,y,z) is
For a tensor field, A, 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,
In Feynman subscript notation,
where the notation ∇B means the subscripted gradient operates on only the factor B. Less general but similar is the Hestenes overdot notation in geometric algebra. The above identity is then expressed as:
where overdots define the scope of the vector derivative. The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant.