Vector calculus identities

The following identities are important in vector calculus:

In the three-dimensional Cartesian coordinate system, the gradient of some function ${\displaystyle f(x,y,z)}$ is given by:

where i, j, k are the standard unit vectors.

The gradient of a tensor field, ${\displaystyle \mathbf {A} }$, of order n, is generally written as

and is a tensor field of order n + 1. In particular, if the tensor field has order 0 (i.e. a scalar), ${\displaystyle \psi }$, the resulting gradient,

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