### Operator notations

### Gradient:

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(x_{1},…,x_{n}), also called a scalar field, the gradient is the vector field:

where e_{i} are orthogonal unit vectors in arbitrary directions.

For a vector field A=(A_{1},…,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:

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.

### Divergence:

In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field F=F_{x}i+F_{y}j+F_{z}k 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,

### Curl:

In Cartesian coordinates, for F=F_{x}i+F_{y}j+F_{z}k 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=(F_{1} F_{2} F_{3})has curl given by:

where ε is the Levi-Civita parity symbol.

### Laplacian:

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,

### Special notations:

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.