# Advanced Mathematics for Engineers and Scientists/The Front Cover

### NomenclatureEdit

 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 OperatorsEdit

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, $\frac{\mathrm d u}{\mathrm d x_i}\,$ 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. $\frac{\partial u}{\partial x_1} \frac{d x_1}{d x_i} + \cdots + \frac{\partial u}{\partial x_n} \frac{d x_n}{d x_i}$
$\nabla u$ 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 SystemsEdit

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

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