Operators are shown applied to the scalar $u(x_{1},x_{2},\cdots ,x_{n})$ or the vector field $\mathbf {v} (x_{1},x_{2},\cdots ,x_{n})=(v_{1},v_{2},\cdots ,v_{n})\,$.

Derivative, total derivative, ${\frac {\mathrm {d} u}{\mathrm {d} x_{i}}}\,$

The rate of change of $u$ with respect to $x_{i}$. If $u$ is multivariate, this derivative will typically depend on the other variables following a path.

Vector that describes the direction and magnitude of the greatest rate of change of a function of more than one variable. The symbol $\nabla$ is called nabla.

Cartesian representations appear in the table above. The $(r,\theta ,\phi )=(\mathrm {distance,azimuth,colatitude} )$ convention is used for spherical coordinates.