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