Partial Differential Equations/The Front Cover

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[edit] 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

[edit] 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 $x i$ , holding the other independent variables constant.	$\lim_{\Delta x_i \to 0} \frac{u(x_1, \cdots, x_i + \Delta x_i, \cdots, x_n) - u}{\Delta x_i}$
$\frac{d u}{d x_i}$	Derivative, total derivative, $\frac{D u}{D x_i}, \frac{\mathrm d u}{\mathrm d x_i}\,$	The rate of change of $u$ with respect to $x i$ , accounting for all variations.	$\frac{\partial u}{\partial x_1} \frac{d x_1}{d x_i} + \cdots + \frac{\partial u}{\partial x_n} \frac{d x_n}{d x_i}$
$\nabla u$	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.	$\nabla(\nabla \cdot \mathbf{\mathbf{v}}) - \nabla \times (\nabla \times \mathbf{\mathbf{v}})$

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