Summary

Vector addition:

$${\bf a}+ {\bf b} \equiv (a_x+b_x, a_y+b_y, a_z+b_z)$$

Vector multiplication:

$$n {\bf a} \equiv (n a_x,n a_y, n a_z)$$

Scalar product:

$${\bf a}\cdot{\bf b} = a_x b_x + a_y b_y + a_z b_z$$

Vector product:

$${\bf a}\times{\bf b} = (a_y b_z-a_z b_y, a_z b_x-a_x b_z, a_x b_y-a_y b_x)$$

Scalar triple product:

$${\bf a}\cdot {\bf b}\times{\bf c} = {\bf a}\times{\bf b}\cdo... ...}\cdot{\bf c}\times{\bf a} = -{\bf b}\cdot{\bf a}\times{\bf c}$$

Vector triple product:

$${\bf a}\times({\bf b}\times{\bf c})= ({\bf a}\cdot{\bf c}) {\bf b} - ({\bf a}\cdot {\bf b}) {\bf c}$$

$$({\bf a}\times{\bf b})\times{\bf c} = ({\bf a}\cdot{\bf c}) {\bf b} -({\bf b} \cdot{\bf c}) {\bf a}$$

Gradient:

$${\bf grad} \phi = \left(\frac{\partial \phi}{\partial x}, ... ... \phi} {\partial y}, \frac{\partial \phi}{\partial z}\right)$$

Divergence:

$${\mit div} {\bf A} = \frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y} +\frac{\partial A_z}{\partial z}$$

Curl:

$${\bf curl} {\bf A} = \left(\frac{\partial A_z}{\partial y}-... ...tial A_y}{\partial x}- \frac{\partial A_x} {\partial y}\right)$$

Gauss' theorem:

$$\oint_S {\bf A}\cdot d{\bf S} = \int_V {\mit div} {\bf A} dV$$

Stokes' theorem:

$$\oint_C {\bf A}\cdot d{\bf l} = \int_S {\bf curl A}\cdot d{\bf S}$$

Del operator:

$$\nabla = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right)$$

$${\bf grad} \phi = \nabla \phi$$

$${\mit div} {\bf A} = \nabla\cdot {\bf A}$$

$${\bf curl A} = \nabla\times{\bf A}$$

Vector identities:

$$\nabla\cdot\nabla \phi = \nabla^2 \phi = \left(\frac{\partia... ...phi}{\partial y^2}+\frac{\partial^2 \phi}{\partial z^2}\right)$$

$$\nabla\cdot\nabla \times {\bf A} = 0$$

$$\nabla\times\nabla\phi = 0$$

$$\nabla^2{\bf A} = \nabla (\nabla\cdot{\bf A})- \nabla\times\nabla\times{\bf A}$$

Other vector identities:

$$\nabla(\phi \psi) =\phi \nabla\psi+\psi \nabla\phi$$

$$\nabla\cdot (\phi {\bf A}) =\phi \nabla\cdot {\bf A} + {\bf A}\cdot \nabla\phi$$

$$\nabla\times(\phi {\bf A}) =\phi \nabla\times{\bf A} +\nabla\phi\times{\bf A}$$

$$\nabla\cdot({\bf A}\times{\bf B}) = {\bf B}\cdot \nabla\times{\bf A} - {\bf A}\cdot \nabla\times{\bf B}$$

$$\nabla\times({\bf A}\times{\bf B}) = {\bf A} (\nabla\cdot {... ... A}) +({\bf B}\cdot\nabla){\bf A}- ({\bf A}\cdot\nabla){\bf B}$$

$$\nabla({\bf A}\cdot{\bf B}) = {\bf A}\times(\nabla\times{\bf... ... +({\bf A}\cdot \nabla){\bf B} + ({\bf B} \cdot \nabla){\bf A}$$

Cylindrical polar coordinates:

$$x = r \cos\theta,   y = r \sin\theta,   z = z,    dV = r dr d\theta dz$$

$$\nabla f = \left(\frac{\partial f}{\partial r}, \frac{1}{r}... ...tial f}{\partial\theta}, \frac{\partial f}{\partial z}\right)$$

$$\nabla\cdot {\bf A} = \frac{1}{r}\frac{\partial (r A_r)}{\pa... ...l A_\theta}{\partial \theta} + \frac{\partial A_z}{\partial z}$$

$$\nabla\times {\bf A}=\left(\frac{1}{r}\frac{\partial A_z}{\p... ...tial r}-\frac{1}{r}\frac{\partial A_r}{\partial \theta}\right)$$

$$\nabla^2 f = \frac{1}{r}\frac{\partial }{\partial r}\!\left(... ...tial^2 f}{\partial\theta^2}+ \frac{\partial^2 f}{\partial z^2}$$

Spherical polar coordinates:

$$x = r \sin\theta \cos\phi,   y = r \sin\theta \sin\phi,   z = r \cos\theta,   dV = r^2 \sin\theta dr d\theta d\phi$$

$$\nabla f = \left(\frac{\partial f}{\partial r}, \frac{1}{r}... ...,\frac{1}{r \sin\theta}\frac{\partial f}{\partial\phi}\right)$$

$$\nabla\cdot{\bf A} = \frac{1}{r^2}\frac{\partial}{\partial r... ...) +\frac{1}{r \sin\theta}\frac{\partial A_\phi}{\partial\phi}$$

$$(\nabla\times{\bf A})_r = \frac{1}{r \sin\theta}\frac{\part... ...-\frac{1}{r \sin\theta}\frac{\partial A_\theta}{\partial\phi}$$

$$(\nabla\times{\bf A})_\theta = \frac{1}{r \sin\theta}\frac... ...artial \phi}-\frac{1}{r}\frac{\partial (r A_\phi)}{\partial r}$$

$$(\nabla\times{\bf A})_z = \frac{1}{r}\frac{\partial (r A_\theta)}{\partial r}-\frac{1}{r}\frac{\partial A_r}{\partial \theta}$$

$$\nabla^2 f = \frac{1}{r^2}\frac{\partial}{\partial r}\!\left... ...\frac{1}{r^2 \sin^2\theta}\frac{\partial^2 f}{\partial\phi^2}$$