# Difference between revisions of "Vector Identities"

Vector Identities
관련코스 다변수 미적분학
소분류 수학, 물리
선행 키워드
연관 키워드

## Contents

### Differential forms

Following identity is often used for deriving wave equation from Maxwell's equation

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

Following two identities are related with the presence of scalar and vector potentials

$\nabla\times(\nabla f) = 0 \quad$ - Curl of any gradient is zero, or gradient is irrotational
$\nabla\cdot(\nabla\times A) = 0 \quad$ - Divergence of any curl is zero, or curl is solenoidal

And others:

$\nabla\times\left(fA\right) = (\nabla f)\times A + f (\nabla\times A)$
$\nabla\cdot (fA) = (\nabla f)\cdot A + f(\nabla\cdot A)$
$\nabla\cdot(A\times B) = B \cdot (\nabla\times A) - A \cdot (\nabla\times B)$
$\nabla(A\cdot B) = (A\cdot\nabla)B + (B\cdot\nabla)A + A\times(\nabla\times B) + B\times(\nabla\times A)$
$\nabla\times(A\times B) = (\nabla\cdot B)A-(\nabla\cdot A)B+(B\cdot\nabla)A-(A\cdot\nabla)B$

### Integral forms

#### Divergence theorem

$\oint_S\mathbf{A}\cdot d\mathbf{a}=\int_V \left(\nabla \cdot \mathbf{A}\right)d\tau \qquad$ used for calculating $\mathbf{E}$ field out of Gauss law in electrostatics

#### Stokes theorem

$\oint_L\mathbf{A}\cdot d\boldsymbol{\ell}=\int_{S}\left(\nabla\times\mathbf{A}\right)\cdot d\mathbf{a} \qquad$ used for calculating $\mathbf{B}$ field from Ampere's law

#### Green's 2nd identity

$\oint_S \left[\left(\psi\nabla\varphi-\varphi\nabla\psi\right)\cdot\hat{\mathbf{n}}\right]da=\,\!\int_S \left[\psi\frac{\partial\varphi}{\partial n}-\varphi\frac{\partial\psi}{\partial n}\right]da = \int_V\left(\psi\nabla^{2}\varphi-\varphi\nabla^{2}\psi\right) d\tau\,\! \qquad$
The derivation of Kirchhoff integral theorem starts from this identity.

#### Others

$-\oint_S \mathbf{A}\times d\mathbf{a}= \int_V\left(\nabla\times\mathbf{A}\right)d\tau \qquad$ used in deriving bound current densities $\mathbf{J}_b$ and $\mathbf{K}_b$ out of magnetization $\mathbf{M}$