Partial derivative ($\partial$)

A partial derivative differentiates a multivariable function with respect to one variable while holding the others fixed.

A mixed partial derivative takes partial derivatives in different variables, for example $f_{xy}={\partial }_y({\partial }_{x}f)$. If both second partial derivatives are continuous in a neighbourhood, their order can be interchanged: $f_{xy}=f_{yx}$. The chain rule transforms mixed partial derivatives under a linear change of variables by composing the corresponding first-order differential operators.

A time derivative is the partial derivative of an evolving field with respect to time while holding its spatial coordinates fixed. For a quantity depending only on time, it is an ordinary derivative.

A spatial derivative is a partial derivative with respect to a spatial coordinate.

The directional derivative of $f$ at $x$ in the direction $v$ isFor a differentiable scalar function on Euclidean space, it equals $v\cdot \nabla f(x)$.

The gradient of a scalar-valued differentiable function is the vector of its first partial derivatives.

A gradient field is a vector field of the form $\nabla f$ for a scalar potential $f$. Its curl vanishes wherever the required second derivatives commute.

For a differentiable scalar function of several variables, the total differential is