The following are important identities involving derivatives and integrals in vector calculus.

For a function ${\displaystyle f(x,y,z)}$ in three-dimensional Cartesian coordinate variables, the gradient is the vector field: $${\displaystyle \operatorname {grad} (f)=\nabla f={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} }$$ where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n variables ${\displaystyle \psi (x_{1},\ldots ,x_{n})}$, also called a scalar field, the gradient is the vector field: $${\displaystyle \nabla \psi ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\end{pmatrix}}\psi ={\frac {\partial \psi }{\partial x_{1}}}\mathbf {e} _{1}+\dots +{\frac {\partial \psi }{\partial x_{n}}}\mathbf {e} _{n}}$$ where ${\displaystyle \mathbf {e} _{i}\,(i=1,2,...,n)}$ are mutually orthogonal unit vectors.

As the name implies, the gradient is proportional to, and points in the direction of, the function's most rapid (positive) change.

For a vector field ${\displaystyle \mathbf {A} =\left(A_{1},\ldots ,A_{n}\right)}$, also called a tensor field of order 1, the gradient or total derivative is the n × n Jacobian matrix: $${\displaystyle \mathbf {J} _{\mathbf {A} }=d\mathbf {A} =(\nabla \!\mathbf {A} )^{\textsf {T}}=\left({\frac {\partial A_{i}}{\partial x_{j}}}\right)_{\!ij}.}$$

For a tensor field ${\displaystyle \mathbf {T} }$ of any order k, the gradient ${\displaystyle \operatorname {grad} (\mathbf {T} )=d\mathbf {T} =(\nabla \mathbf {T} )^{\textsf {T}}}$ is a tensor field of order k + 1.

For a tensor field ${\displaystyle \mathbf {T} }$ of order k > 0, the tensor field ${\displaystyle \nabla \mathbf {T} }$ of order k + 1 is defined by the recursive relation $${\displaystyle (\nabla \mathbf {T} )\cdot \mathbf {C} =\nabla (\mathbf {T} \cdot \mathbf {C} )}$$ where ${\displaystyle \mathbf {C} }$ is an arbitrary constant vector.

In Cartesian coordinates, the divergence of a continuously differentiable vector field ${\displaystyle \mathbf {F} =F_{x}\mathbf {i} +F_{y}\mathbf {j} +F_{z}\mathbf {k} }$ is the scalar-valued function: $${\displaystyle {\begin{aligned}\operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} &={\begin{pmatrix}{\dfrac {\partial }{\partial x}},\ {\dfrac {\partial }{\partial y}},\ {\dfrac {\partial }{\partial z}}\end{pmatrix}}\cdot {\begin{pmatrix}F_{x},\ F_{y},\ F_{z}\end{pmatrix}}\\[1ex]&={\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}.\end{aligned}}}$$

As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.