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# 物理数学I ベクトル解析/付録

## 定義式

3次元デカルト座標での定義式。

${\displaystyle \nabla ={\begin{pmatrix}{\partial /\partial x}\\{\partial /\partial y}\\{\partial /\partial z}\end{pmatrix}}}$

${\displaystyle \mathrm {grad} \phi =\nabla \phi ={\begin{pmatrix}{\partial \phi /\partial x}\\{\partial \phi /\partial y}\\{\partial \phi /\partial z}\end{pmatrix}}}$

${\displaystyle \mathrm {div} \mathbf {F} =\nabla \cdot \mathbf {F} =\partial F_{x}/\partial x+\partial F_{y}/\partial y+\partial F_{z}/\partial z}$

${\displaystyle \mathrm {curl} \mathbf {F} =\mathrm {rot} \mathbf {F} =\nabla \times \mathbf {F} ={\begin{pmatrix}{\partial F_{z}/\partial y}-{\partial F_{y}/\partial z}\\{\partial F_{x}/\partial z}-{\partial F_{z}/\partial x}\\{\partial F_{y}/\partial x}-{\partial F_{x}/\partial y}\end{pmatrix}}}$

${\displaystyle \Delta =\nabla ^{2}={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}}$

${\displaystyle \Delta \phi =\nabla ^{2}\phi =\nabla \cdot (\nabla \phi )=\mathrm {div} (\mathrm {grad} \phi )}$

## 重要な公式

${\displaystyle \operatorname {div} (a\mathbf {F} +b\mathbf {G} )=a\;\operatorname {div} (\mathbf {F} )+b\;\operatorname {div} (\mathbf {G} )}$

${\displaystyle \operatorname {div} (\phi \mathbf {F} )=\operatorname {grad} (\phi )\cdot \mathbf {F} +\phi \;\operatorname {div} (\mathbf {F} )}$

${\displaystyle \nabla \cdot (\phi \mathbf {F} )=(\nabla \phi )\cdot \mathbf {F} +\phi \;(\nabla \cdot \mathbf {F} )}$

${\displaystyle \operatorname {div} (\mathbf {F} \times \mathbf {G} )=\operatorname {curl} (\mathbf {F} )\cdot \mathbf {G} \;-\;\mathbf {F} \cdot \operatorname {curl} (\mathbf {G} )}$

### div(curl F)

${\displaystyle \operatorname {div} (\operatorname {curl} \mathbf {F} )=\operatorname {div} (\nabla \times \mathbf {F} )=\operatorname {curl} (\nabla )\cdot \mathbf {F} -\nabla \cdot \operatorname {curl} (\mathbf {F} )}$

ここで ${\displaystyle \left[\operatorname {curl} (\nabla )\right]_{x}={\frac {\partial ^{2}}{\partial z\partial y}}-{\frac {\partial ^{2}}{\partial y\partial z}}=0}$ (演算対象の関数が連続でなめらかな場合) であるので

${\displaystyle \operatorname {div} (\operatorname {curl} \mathbf {F} )=-\nabla \cdot \operatorname {curl} (\mathbf {F} )=-\operatorname {div} (\operatorname {curl} \mathbf {F} )}$

### curl(curl F)

${\displaystyle \operatorname {curl} (\operatorname {curl} (\mathbf {F} ))=-\Delta \mathbf {F} +\operatorname {grad} (\operatorname {div} \mathbf {F} )}$

x成分をとって証明する。

${\displaystyle \left[\operatorname {curl} (\operatorname {curl} (\mathbf {F} ))\right]_{x}=\left[\nabla \times (\nabla \times \mathbf {F} )\right]_{x}={\frac {\partial }{\partial y}}\left[\nabla \times \mathbf {F} \right]_{z}-{\frac {\partial }{\partial z}}\left[\nabla \times \mathbf {F} \right]_{y}}$ ${\displaystyle ={\frac {\partial }{\partial y}}({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}})-{\frac {\partial }{\partial z}}({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}})}$ ${\displaystyle =-({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}})F_{x}+{\frac {\partial }{\partial x}}({\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}})}$ ${\displaystyle =-\Delta F_{x}+{\frac {\partial }{\partial x}}\operatorname {div} \mathbf {F} =\left[-\Delta \mathbf {F} +\operatorname {grad} (\operatorname {div} \mathbf {F} )\right]_{x}}$