# Advanced Mathematics for Engineers and Scientists/The Front Cover

(Redirected from Partial Differential Equations/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 ${\displaystyle u(x_{1},x_{2},\cdots ,x_{n})}$  or the vector field ${\displaystyle \mathbf {v} (x_{1},x_{2},\cdots ,x_{n})=(v_{1},v_{2},\cdots ,v_{n})\,}$ .

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

## 3D Operators in Different Coordinate SystemsEdit

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

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