2014/03/27 | Improve Society

If the partial derivatives of $\bold{A}$ , $\bold{B}$ , $U$ and $V$ are assumed to exist, then

$\displaystyle \nabla(U + V) = \nabla U + \nabla V$ or $grad(U + V) = grad U + grad V$

$\displaystyle \nabla\cdot(\bold{A} + \bold{B}) = \nabla\cdot\bold{A} + \nabla\cdot\bold{B}$ or $div(\bold{A} + \bold{B}) = div\bold{A} + div\bold{B}$

$\displaystyle \nabla\times (\bold{A} + \bold{B}) = \nabla\times\bold{A} + \nabla\times\bold{B}$ or $curl(\bold{A} + \bold{B}) = curl\bold{A} + curl\bold{B}$

$\displaystyle \nabla\cdot(U\bold{A}) = (\nabla U)\cdot\bold{A} + U(\nabla\cdot\bold{A})$

$\displaystyle \nabla\times(U\bold{A}) = (\nabla U)\times\bold{A} + U(\nabla\times\bold{A})$

$\displaystyle \nabla\cdot(\bold{A}\times\bold{B}) = \bold{B}\cdot(\nabla\times\bold{A}) - \bold{A}\cdot(\nabla\times\bold{B})$

$\displaystyle \nabla\times(\bold{A}\times\bold{B}) = (\bold{B}\cdot\nabla)\bold{A} - \bold{B}(\nabla\cdot\bold{A}) - (\bold{A}\cdot\nabla)\bold{B} + \bold{A}(\nabla\cdot\bold{B})$

$\displaystyle \nabla(\bold{A}\cdot\bold{B}) = (\bold{B}\cdot\nabla)\bold{A} + (\bold{A}\cdot\nabla)\bold{B} + \bold{B}\times(\nabla\times\bold{A}) + \bold{A}\times(\nabla\times\bold{B})$

$\displaystyle \nabla\cdot(\nabla U) \equiv \nabla^2U \equiv \frac{\partial^2U}{\partial x^2} + \frac{\partial^2U}{\partial y^2} + \frac{\partial^2U}{\partial z^2}$ is called Laplacian of $U$ and
$\displaystyle \nabla^2 \equiv \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ is called Laplacian operator.

$\displaystyle \nabla\times(\nabla U) = 0$ . The curl of the gradient of $U$ is zero.

$\displaystyle \nabla\cdot(\nabla\times\bold{A}) = 0$ . The divergence of the curl of $\bold{A}$ is zero.

$\displaystyle \nabla \times (\nabla \times \bold{A}) = \nabla (\nabla \cdot \bold{A}) - \nabla^2\bold{A}$

　仮に $\bold{A}$ , $\bold{B}$ , $U$ および $V$ の偏微分が存在すると仮定すると

$\displaystyle \nabla(U + V) = \nabla U + \nabla V$ または $grad(U + V) = grad U + grad V$
$\displaystyle \nabla\cdot(\bold{A} + \bold{B}) = \nabla\cdot\bold{A} + \nabla\cdot\bold{B}$ または $div(\bold{A} + \bold{B}) = div\bold{A} + div\bold{B}$
$\displaystyle \nabla\times (\bold{A} + \bold{B}) = \nabla\times\bold{A} + \nabla\times\bold{B}$ または $curl(\bold{A} + \bold{B}) = curl\bold{A} + curl\bold{B}$
$\displaystyle \nabla\cdot(U\bold{A}) = (\nabla U)\cdot\bold{A} + U(\nabla\cdot\bold{A})$
$\displaystyle \nabla\times(U\bold{A}) = (\nabla U)\times\bold{A} + U(\nabla\times\bold{A})$
$\displaystyle \nabla\cdot(\bold{A}\times\bold{B}) = \bold{B}\cdot(\nabla\times\bold{A}) - \bold{A}\cdot(\nabla\times\bold{B})$
$\displaystyle \nabla\times(\bold{A}\times\bold{B}) = (\bold{B}\cdot\nabla)\bold{A} - \bold{B}(\nabla\cdot\bold{A}) - (\bold{A}\cdot\nabla)\bold{B} + \bold{A}(\nabla\cdot\bold{B})$
$\displaystyle \nabla(\bold{A}\cdot\bold{B}) = (\bold{B}\cdot\nabla)\bold{A} + (\bold{A}\cdot\nabla)\bold{B} + \bold{B}\times(\nabla\times\bold{A}) + \bold{A}\times(\nabla\times\bold{B})$
$\displaystyle \nabla\cdot(\nabla U) \equiv \nabla^2U \equiv \frac{\partial^2U}{\partial x^2} + \frac{\partial^2U}{\partial y^2} + \frac{\partial^2U}{\partial z^2}$ は $U$ の ラプラス と呼び，
$\displaystyle \nabla^2 \equiv \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ は ラプラス演算子 と呼ぶ．
$\displaystyle \nabla\times(\nabla U) = 0$ . $U$ の勾配の回転はゼロである．
$\displaystyle \nabla\cdot(\nabla\times\bold{A}) = 0$ . $\bold{A}$ の回転の発散はゼロである．
$\displaystyle \nabla \times (\nabla \times \bold{A}) = \nabla (\nabla \cdot \bold{A}) - \nabla^2\bold{A}$

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日: 2014年3月27日

Formulas involving ∇

∇を有する式