Vector Identities

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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

[math]\nabla\times\left(\nabla\times A\right)=\nabla\left(\nabla\cdot A\right)-{\nabla}^2 A[/math]

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

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

And others:

[math]\nabla\times\left(fA\right) = (\nabla f)\times A + f (\nabla\times A) [/math]
[math]\nabla\cdot (fA) = (\nabla f)\cdot A + f(\nabla\cdot A) [/math]
[math]\nabla\cdot(A\times B) = B \cdot (\nabla\times A) - A \cdot (\nabla\times B) [/math]
[math]\nabla(A\cdot B) = (A\cdot\nabla)B + (B\cdot\nabla)A + A\times(\nabla\times B) + B\times(\nabla\times A)[/math]
[math]\nabla\times(A\times B) = (\nabla\cdot B)A-(\nabla\cdot A)B+(B\cdot\nabla)A-(A\cdot\nabla)B[/math]

Integral forms

Divergence theorem

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

Stokes theorem

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

Green's 2nd identity

[math]\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[/math]
The derivation of Kirchhoff integral theorem starts from this identity.

Others

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