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관련코스	다변수 미적분학
소분류	수학, 물리
선행 키워드
연관 키워드

Definition

[math] \nabla\times\mathbf{A} = \frac{1}{h_1h_2h_3} \mathbf{e}_i \epsilon_{ijk} h_i \frac{\partial (h_k A_k)}{\partial q^j} [/math]

where [math]\epsilon_{ijk}[/math] is the Levi-Civita symbol.

Or, more explicitly,

[math] \nabla\times\mathbf{A} = \frac{1}{h_1h_2h_3} \left| \begin{array}{ccc} h_1\textbf{e}_1 & h_2\textbf{e}_2 & h_3\textbf{e}_3 \\ {\partial\over\partial q_1} & {\partial\over\partial q_2} &{\partial\over\partial q_3} \\ h_1A_1 & h_2A_2 & h_3A_3 \end{array} \right|[/math]

Cartesian coordinates

[math]\nabla\times\mathbf{A} = \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) \hat{\mathbf x} + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) \hat{\mathbf y} + \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) \hat{\mathbf z} [/math]

Cylindrical coordinates

[math]\nabla\times\mathbf{A} = \left( \frac{1}{\rho} \frac{\partial A_z}{\partial \varphi} - \frac{\partial A_\varphi}{\partial z} \right) \hat{\boldsymbol \rho} + \left( \frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho} \right) \hat{\boldsymbol \varphi} + \frac{1}{\rho} \left( \frac{\partial \left( \rho A_\varphi\right) }{\partial \rho} - \frac{\partial A_\rho}{\partial \varphi} \right) \hat{\mathbf z}[/math]

Spherical coordinates

[math]\nabla\times\mathbf{A} = \frac{1}{r\sin\theta} \left(\frac{\partial}{\partial \theta} \left(A_\varphi\sin\theta \right) - \frac{\partial A_\theta}{\partial \varphi} \right) \hat{\mathbf r} + \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \varphi} - \frac{\partial}{\partial r} \left( r A_\varphi \right) \right) \hat{\boldsymbol \theta} {}+ \frac{1}{r} \left( \frac{\partial}{\partial r} \left( r A_{\theta} \right) - \frac{\partial A_r}{\partial \theta} \right) \hat{\boldsymbol \varphi}[/math]