Difference between revisions of "Vector Identities"

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

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 $\mathbf{M}$