Advanced Mathematics for Engineers and Scientists/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 ${\displaystyle u(x_{1},x_{2},\cdots ,x_{n})}$ 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 Systems

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