Partial Differential Equations/The Front Cover

Nomenclature

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

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 xi, holding the other independent variables constant.	$\lim_{\Delta x_i \to 0} \frac{u(x_1, \cdots, x_i + \Delta x_i, \cdots, x_n) - u}{\Delta x_i}$
$\frac{d u}{d x_i}$	Derivative, total derivative, $\frac{\mathrm d u}{\mathrm d x_i}\,$	The rate of change of u with respect to xi. If u is multivariate, this derivative will typically depend on the other variables following a path.	$\frac{\partial u}{\partial x_1} \frac{d x_1}{d x_i} + \cdots + \frac{\partial u}{\partial x_n} \frac{d x_n}{d x_i}$
$\nabla u$	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

Cartesian representations appear in the table above. The (r,θ,ϕ) = (distance,azimuth,colatitude) convention is used for spherical coordinates.

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