Advanced Mathematics for Engineers and Scientists/The Front Cover

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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[edit]

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 Systems[edit]

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)\,