Nomenclature Edit
ODE
Ordinary Differential Equation
PDE
Partial Differential Equation
BC
Boundary Condition
IVP
Initial Value Problem
BVP
Boundary Value Problem
IBVP
Initial Boundary Value Problem

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

Notation
Common names and other notation
Description and notes
Definition in Cartesian coordinates
${\frac {\partial u}{\partial x_{i}}}$
Partial derivative, $u_{x_{i}},\ \partial _{x_{i}}u\,$
The rate of change of $u$ with respect to $x_{i}$ , holding the other independent variables constant.
$\lim _{\Delta x_{i}\to 0}{\frac {u(x_{1},\cdots ,x_{i}+\Delta x_{i},\cdots ,x_{n})-u}{\Delta x_{i}}}$
${\frac {du}{dx_{i}}}$
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.
${\frac {\partial u}{\partial x_{1}}}{\frac {dx_{1}}{dx_{i}}}+\cdots +{\frac {\partial u}{\partial x_{n}}}{\frac {dx_{n}}{dx_{i}}}$
$\nabla u$
Gradient, del operator, $\mathrm {grad} \ u\,$
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 .
$\left({\frac {\partial u}{\partial x_{1}}},\cdots ,{\frac {\partial u}{\partial x_{n}}}\right)$
$\nabla ^{2}u$
Laplacian, Scalar Laplacian, Laplace operator, $\Delta u,\ (\nabla \cdot \nabla )u\,$
A measure of the concavity of $u$ , equivalently a comparison of the value of $u$ at some point to neighboring values.
${\frac {\partial ^{2}u}{\partial x_{1}^{2}}}+\cdots +{\frac {\partial ^{2}u}{\partial x_{n}^{2}}}$
$\nabla \cdot \mathbf {v}$
Divergence, $\mathrm {div} \ \mathbf {v} \,$
A measure of "generation", in other words how much the vector field acts as a source or sink at a point.
${\frac {\partial v_{1}}{\partial x_{1}}}+\cdots +{\frac {\partial v_{n}}{\partial x_{n}}}$
$\nabla \times \mathbf {v}$
Curl, rotor, circulation density, $\mathrm {curl} \ \mathbf {v} ,\ \mathrm {rot} \ \mathbf {v} \,$
A vector that describes the rate of rotation of a (normally 3D) vector field and the corresponding axis of rotation.
$\left({\frac {\partial v_{3}}{\partial x_{2}}}-{\frac {\partial v_{2}}{\partial x_{3}}},{\frac {\partial v_{1}}{\partial x_{3}}}-{\frac {\partial v_{3}}{\partial x_{1}}},{\frac {\partial v_{2}}{\partial x_{1}}}-{\frac {\partial v_{1}}{\partial x_{2}}}\right)$
$\nabla ^{2}\mathbf {v}$
Vector Laplacian
Similar to the (scalar) Laplacian. Note however, that it is generally not equal to the element-by-element Laplacian of a vector.
$\nabla (\nabla \cdot \mathbf {\mathbf {v} } )-\nabla \times (\nabla \times \mathbf {\mathbf {v} } )$

3D Operators in Different Coordinate Systems Edit
Cartesian representations appear in the table above. The $(r,\theta ,\phi )=(\mathrm {distance,azimuth,colatitude} )$ convention is used for spherical coordinates.

Operator
Cylindrical
Spherical
$\nabla u$
$\left({\frac {\partial u}{\partial r}},{\frac {1}{r}}{\frac {\partial u}{\partial \theta }},{\frac {\partial u}{\partial z}}\right)\,$
$\left({\frac {\partial u}{\partial r}},{\frac {1}{r\sin(\phi )}}{\frac {\partial u}{\partial \theta }},{\frac {1}{r}}{\frac {\partial u}{\partial \phi }}\right)\,$
$\nabla ^{2}u$
${\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial u}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}u}{\partial \theta ^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}\,$
${\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial u}{\partial r}}\right)+{\frac {1}{r^{2}\sin(\phi )}}{\frac {\partial ^{2}u}{\partial \theta ^{2}}}+{\frac {1}{r^{2}\sin(\phi )}}{\frac {\partial }{\partial \phi }}\left(\sin(\phi ){\frac {\partial u}{\partial \phi }}\right)\,$
$\nabla \cdot \mathbf {v}$
${\frac {1}{r}}{\frac {\partial }{\partial r}}\left(rv_{r}\right)+{\frac {1}{r}}{\frac {\partial v_{\theta }}{\partial \theta }}+{\frac {\partial v_{z}}{\partial z}}\,$
${\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}v_{r}\right)+{\frac {1}{r\sin(\phi )}}{\frac {\partial v_{\theta }}{\partial \theta }}+{\frac {1}{r\sin(\phi )}}{\frac {\partial }{\partial \phi }}\left(\sin(\phi )v_{\phi }\right)\,$