Divergence

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Divergence
관련코스 다변수 미적분학
소분류 수학, 물리
선행 키워드
연관 키워드

Definition

$\nabla\cdot\mathbf{A} = \cfrac{1}{\prod_j h_j} \frac{\partial }{\partial q^i}(A^i\prod_{j\ne i} h_j)$
Or, more explicitly in 3-dim,
$\nabla\cdot\mathbf{A} = \cfrac{1}{h_1h_2h_3} \left[ {\partial\over\partial q_1}(A_1h_2h_3) + {\partial\over\partial q_2}(A_2h_3h_1) + {\partial\over\partial q_3}(A_3h_1h_2) \right]$

Cartesian coordinates

$\nabla\cdot\mathbf{A} = {\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}$

Cylindrical coordinates

$\nabla\cdot\mathbf{A} = {1 \over \rho}{\partial \left( \rho A_\rho \right) \over \partial \rho} + {1 \over \rho}{\partial A_\varphi \over \partial \varphi} + {\partial A_z \over \partial z}$

Spherical coordinates

$\nabla\cdot\mathbf{A} = {1 \over r^2}{\partial \left( r^2 A_r \right) \over \partial r} + {1 \over r\sin\theta}{\partial \over \partial \theta} \left( A_\theta\sin\theta \right) + {1 \over r\sin\theta}{\partial A_\varphi \over \partial \varphi}$