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高校〜大学基礎の数学用語.公式.例

\( \displaystyle {\rm grad}\hspace{2px}(\phi+\psi)={\rm grad}\hspace{2px}\phi+{\rm grad}\hspace{2px}\psi \)
\( \displaystyle \nabla(\phi+\psi)=\nabla\phi+\nabla\psi \)

≪定義通りに計算すれば≫
\( \displaystyle {\rm grad}\hspace{2px}(\phi+\psi)=\Big(\frac{\partial}{\partial x}(\phi+\psi),\hspace{2px}\frac{\partial}{\partial y}(\phi+\psi),\hspace{2px}\frac{\partial}{\partial z}(\phi+\psi)\Big) \)
\( \displaystyle =(\frac{\partial\phi}{\partial x}+\frac{\partial\psi}{\partial x}, \) \( \displaystyle \hspace{2px}\frac{\partial\phi}{\partial y}+\frac{\partial\psi}{\partial y},\hspace{2px}\frac{\partial\phi}{\partial z}+\frac{\partial\psi}{\partial z}\Big) \)
\( \displaystyle =(\frac{\partial\phi}{\partial x},\hspace{2px}\frac{\partial\phi}{\partial y},\hspace{2px}\frac{\partial\phi}{\partial z}) \) \( \displaystyle +(\frac{\partial\psi}{\partial x},\hspace{2px}\frac{\partial\psi}{\partial y},\hspace{2px}\frac{\partial\psi}{\partial z}) \)
\( \displaystyle ={\rm grad}\hspace{2px}\phi+{\rm grad}\hspace{2px}\psi \)
≪ベクトルの内積・外積で表せば≫
\( \displaystyle {\rm grad}\hspace{2px}(\phi+\psi)=\nabla(\phi+\psi)=\nabla\phi+\nabla\psi \)

\( \displaystyle {\rm grad}\hspace{2px}(\phi\psi)=\phi\hspace{4px}{\rm grad}\hspace{2px}\psi+\psi\hspace{4px}{\rm grad}\hspace{2px}\phi \)
\( \displaystyle \nabla(\phi\psi)=\phi\nabla\psi+\psi\nabla\phi \)

≪定義通りに計算すれば≫
\( \displaystyle {\rm grad}\hspace{2px}(\phi\psi)=\Big(\frac{\partial}{\partial x}(\phi\psi),\hspace{2px}\frac{\partial}{\partial y}(\phi\psi),\hspace{2px}\frac{\partial}{\partial z}(\phi\psi)\Big) \)
積の微分法：\( \displaystyle \frac{\partial}{\partial x}(fg)=\frac{\partial f}{\partial x}g+ f\frac{\partial g}{\partial x} \) 等により
\( \displaystyle \frac{\partial}{\partial x}(\phi\psi)=\frac{\partial\phi}{\partial x}\psi+\phi\frac{\partial\psi}{\partial x} \)
\( \displaystyle \frac{\partial}{\partial y}(\phi\psi)=\frac{\partial\phi}{\partial y}\psi+\phi\frac{\partial\psi}{\partial y} \)
\( \displaystyle \frac{\partial}{\partial z}(\phi\psi)=\frac{\partial\phi}{\partial z}\psi+\phi\frac{\partial\psi}{\partial z} \)

したがって
\( \displaystyle \Big(\frac{\partial\phi}{\partial x}\psi+\phi\frac{\partial\psi}{\partial x},\hspace{4px} \) \( \displaystyle \frac{\partial\phi}{\partial y}\psi+\phi\frac{\partial\psi}{\partial y},\hspace{4px} \) \( \displaystyle \frac{\partial\phi}{\partial z}\psi+\phi\frac{\partial\psi}{\partial z}\Big) \)
\( \displaystyle =\Big(\frac{\partial\phi}{\partial x},\hspace{4px}\frac{\partial\phi}{\partial y},\hspace{4px}\frac{\partial\phi}{\partial z}\Big)\psi \) \( \displaystyle +\phi\Big(\frac{\partial\psi}{\partial x},\hspace{4px}\frac{\partial\psi}{\partial y},\hspace{4px}\frac{\partial\psi}{\partial z}\Big) \)
\( \displaystyle =\phi\Big(\frac{\partial\psi}{\partial x},\hspace{4px}\frac{\partial\psi}{\partial y},\hspace{4px}\frac{\partial\psi}{\partial z}\Big) \) \( \displaystyle +\psi\Big(\frac{\partial\phi}{\partial x},\hspace{4px}\frac{\partial\phi}{\partial y},\hspace{4px}\frac{\partial\phi}{\partial z}\Big) \)
\( \displaystyle =\phi\hspace{4px}{\rm grad}\hspace{2px}\psi+\psi\hspace{4px}{\rm grad}\hspace{2px}\phi \)
≪ベクトルの内積・外積で表せば≫
ベクトルの実数倍（定数倍）の公式では，\( \displaystyle \vec{a}k=k\vec{a} \)であるが，ナブラ
\( \displaystyle \nabla=(\frac{\partial}{\partial x},\hspace{2px}\frac{\partial}{\partial y},\hspace{2px}\frac{\partial}{\partial z}) \)
は微分演算子であるからスカラー関数\( \displaystyle \phi \)と左右入れ換えることはできない． \( \displaystyle \nabla\phi\leftarrow\circ \)， \( \displaystyle \phi\nabla\leftarrow\times \)
　また，ベクトルの実数倍（定数倍）の公式では， \( \displaystyle \vec{a}(kl)=kl\vec{a} \)であるが，上記のようにナブラは微分演算子であるから，積の微分法が成り立ち， \( \displaystyle \nabla(\phi\psi)=\phi\psi\nabla \)などとは変形できない．結論から言えば， \( \displaystyle {\rm grad}\hspace{2px}(\phi\psi)=\phi\hspace{4px}{\rm grad}\hspace{2px}\psi+\psi\hspace{4px}{\rm grad}\hspace{2px}\phi \)をナブラで表すと， \( \displaystyle \nabla(\phi\psi)=\phi\nabla\psi+\psi\nabla\phi \)となる．

\( \displaystyle {\rm grad}\hspace{4px}{\rm div}\hspace{4px}\vec{V}={\rm rot}\hspace{4px}{\rm rot}\hspace{4px}\vec{V}+\Delta\vec{V} \)
\( \displaystyle \nabla(\nabla\cdot\vec{V})=\nabla\times(\nabla\times\vec{V})+\Delta\vec{V} \)

\( \displaystyle {\rm div}\hspace{4px}{\rm grad}\hspace{4px}\phi={\rm \Delta}\phi \)
\( \displaystyle \nabla\cdot\nabla\phi=\Delta\phi \)

≪定義通りに計算すれば≫
\( \displaystyle {\rm grad}\hspace{4px}\phi=(\frac{\partial\phi}{\partial x},\hspace{2px}\frac{\partial\phi}{\partial y},\hspace{2px}\frac{\partial\phi}{\partial z}) \)
\( \displaystyle {\rm div}\hspace{4px}{\rm grad}\hspace{4px}\phi=\frac{\partial}{\partial x}(\frac{\partial\phi}{\partial x}) \) \( \displaystyle +\frac{\partial}{\partial y}(\frac{\partial\phi}{\partial y})+\frac{\partial}{\partial z}(\frac{\partial\phi}{\partial z}) \)
\( \displaystyle =\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2} \)
≪ベクトルの内積・外積で表せば≫
\( \displaystyle {\rm grad}\hspace{4px}\phi=\nabla\phi \)
\( \displaystyle {\rm div}\hspace{4px}{\rm grad}\hspace{4px}\phi=\nabla\cdot\nabla\phi=\Delta\phi \)

\( \displaystyle {\rm div}(\vec{V}+\vec{W})={\rm div}\vec{V}+\rm{div}\vec{W} \)
\( \displaystyle \nabla\cdot(\vec{V}+\vec{W})=\nabla\cdot\vec{V}+\nabla\cdot\vec{W} \)

≪定義通りに計算すれば≫
\( \displaystyle \vec{V}+\vec{W}=(V_x,V_y,V_z)+(W_x,W_y,W_z) \)
\( \displaystyle =(V_x+ W_x,V_y+ W_y,V_z+ W_z) \)
\( \displaystyle {\rm div}(\vec{V}+\vec{W})=\frac{\partial}{\partial x}(V_x+ W_x)+\frac{\partial}{\partial y}(V_y+ W_y) \)
\( \displaystyle +\frac{\partial}{\partial z}(V_z+ W_z) \)
\( \displaystyle =(\frac{\partial V_x}{\partial x}+\frac{\partial V_y}{\partial y}+\frac{\partial V_z}{\partial z})+(\frac{\partial W_x}{\partial x}+\frac{\partial W_y}{\partial y}+\frac{\partial W_z}{\partial z}) \)
\( \displaystyle ={\rm div}\vec{V}+\rm{div}\vec{W} \)
≪ベクトルの内積・外積で表せば≫
\( \displaystyle {\rm div}(\vec{V}+\vec{W})=\nabla\cdot(\vec{V}+\vec{W})=\nabla\cdot\vec{V}+\nabla\cdot\vec{W} \)

\( \displaystyle {\rm div}(\phi\vec{V})=\phi\hspace{4px}{\rm div}\vec{V}+{\rm grad}\phi\cdot\vec{V} \)
\( \displaystyle \nabla\cdot(\phi\vec{V})=\nabla\phi\cdot\vec{V}+\phi\nabla\cdot\vec{V} \)

≪定義通りに計算すれば≫
\( \displaystyle {\rm div}(\phi\vec{V})=\frac{\partial}{\partial x}(\phi V_x)+\frac{\partial}{\partial y}(\phi V_y)+\frac{\partial}{\partial z}(\phi V_z) \)
積の微分法：\( \displaystyle \frac{\partial}{\partial x}(fg)=\frac{\partial f}{\partial x}g+ f\frac{\partial g}{\partial x} \)
などにより
\( \displaystyle \frac{\partial}{\partial x}(\phi V_x)=\frac{\partial\phi}{\partial x}V_x+\phi\frac{\partial V_x}{\partial x} \)
\( \displaystyle \frac{\partial}{\partial y}(\phi V_y)=\frac{\partial\phi}{\partial y}V_y+\phi\frac{\partial V_y}{\partial y} \)
\( \displaystyle \underline{+\hspace{5px}\big)\frac{\partial}{\partial z}(\phi V_z)=\frac{\partial\phi}{\partial z}V_z+\phi\frac{\partial V_z}{\partial z}} \)
右辺は\( \displaystyle {\rm grad}\phi\cdot\vec{V}+\phi\hspace{4px}{\rm div}\vec{V} \)
だから等しい
≪ベクトルの内積・外積で表せば≫
\( \displaystyle {\rm div}(\phi\vec{V})=\nabla\cdot(\phi\vec{V}) \)
\( \displaystyle =\nabla\phi\cdot\vec{V}+\phi\nabla\cdot\vec{V} \)
※積の微分が含まれているから，ベクトルの公式\( \displaystyle \vec{a}\cdot(k\vec{b})=k\vec{a}\cdot\vec{b} \)のままではない

\( \displaystyle {\rm div}(\phi\hspace{4px}{\rm grad}\psi)={\rm grad}\phi\cdot{\rm grad}\psi+\phi\Delta \psi \)
\( \displaystyle \nabla\cdot(\phi\nabla\psi)=\nabla\phi\cdot\nabla\psi+\phi\Delta \psi \)

≪ベクトルの内積・外積で表せば≫
\( \displaystyle \nabla\psi=\vec{e_x}\frac{\partial\psi}{\partial x}+\vec{e_y}\frac{\partial\psi}{\partial y}+\vec{e_z}\frac{\partial\psi}{\partial z} \)
\( \displaystyle \phi\hspace{2px}\nabla\psi=\vec{e_x}\hspace{2px}\phi\hspace{2px}\frac{\partial\psi}{\partial x} \) \( \displaystyle +\vec{e_y}\hspace{2px}\phi\hspace{2px}\frac{\partial\psi}{\partial y}+\vec{e_z}\hspace{2px}\phi\hspace{2px}\frac{\partial\psi}{\partial z} \)

\( \displaystyle \nabla\cdot(\phi\hspace{2px}\nabla\psi) \)
\( \displaystyle =(\vec{e_x}\frac{\partial}{\partial x}+\vec{e_y}\frac{\partial}{\partial y}+\vec{e_z}\frac{\partial}{\partial z}) \) \( \displaystyle \cdot(\vec{e_x}\hspace{2px}\phi\hspace{2px}\frac{\partial\psi}{\partial x} \) \( \displaystyle +\vec{e_y}\hspace{2px}\phi\hspace{2px}\frac{\partial\psi}{\partial y}+\vec{e_z}\hspace{2px}\phi\hspace{2px}\frac{\partial\psi}{\partial z}\Big) \)
\( \displaystyle =\frac{\partial}{\partial x}(\phi\hspace{2px}\frac{\partial\psi}{\partial x}) \) \( \displaystyle +\frac{\partial}{\partial y}(\phi\hspace{2px}\frac{\partial\psi}{\partial y})+\frac{\partial}{\partial z}(\phi\hspace{2px}\frac{\partial\psi}{\partial z}) \) \( \displaystyle =\frac{\partial\phi}{\partial x}\frac{\partial\psi}{\partial x}+\phi\frac{\partial^2\psi}{\partial x^2} \) \( \displaystyle +\frac{\partial\phi}{\partial y}\frac{\partial\psi}{\partial y}+\phi\frac{\partial^2\psi}{\partial y^2} \) \( \displaystyle +\frac{\partial\phi}{\partial z}\frac{\partial\psi}{\partial z}+\phi\frac{\partial^2\psi}{\partial z^2} \)
\( \displaystyle =\frac{\partial\phi}{\partial x}\frac{\partial\psi}{\partial x}+\frac{\partial\phi}{\partial y}\frac{\partial\psi}{\partial y}+\frac{\partial\phi}{\partial z}\frac{\partial\psi}{\partial z} \) \( \displaystyle +\phi\frac{\partial^2\psi}{\partial x^2}+\phi\frac{\partial^2\psi}{\partial y^2}+\phi\frac{\partial^2\psi}{\partial z^2} \)
\( \displaystyle =\nabla\phi\cdot\nabla\psi+\phi\Delta \psi \)

\( \displaystyle {\rm div}(\vec{V}\times\vec{W})=\vec{W}\cdot\rm{rot}\vec{V}-\vec{V}\cdot\rm{rot}\vec{W} \)
\( \displaystyle \nabla\cdot(\vec{V}\times\vec{W})=\vec{W}\cdot(\nabla\times\vec{V})-\vec{V}\cdot(\nabla\times\vec{W}) \)

≪定義通りに計算すれば≫
\( \displaystyle \vec{V}\times\vec{W}=\begin{vmatrix}\vec{e_x}&\vec{e_y}&\vec{e_z}\\ V_x& V_y& V_z\\ W_x& W_y& W_z\end{vmatrix} \)
\( \displaystyle =(V_yW_z-V_zW_y,\hspace{5px}V_zW_x-V_xW_z,\hspace{5px}V_xW_y-V_yW_x) \)
\( \displaystyle {\rm div}(\vec{V}\times\vec{W}) \)
\( \displaystyle =\frac{\partial}{\partial x}(V_yW_z-V_zW_y) \)
\( \displaystyle +\frac{\partial}{\partial y}(V_zW_x-V_xW_z) \)
\( \displaystyle +\frac{\partial}{\partial z}(V_xW_y-V_yW_x) \)
積の微分法：\( \displaystyle \frac{\partial}{\partial x}(V_yW_z)=\frac{\partial V_y}{\partial x}W_z+ V_y\frac{\partial W_z}{\partial x} \)
などにより
\( \displaystyle \frac{\partial V_y}{\partial x}W_z \) \( \displaystyle + V_y\frac{\partial W_z}{\partial x} \) \( \displaystyle -\frac{\partial V_z}{\partial x}W_y \) \( \displaystyle -V_z\frac{\partial W_y}{\partial x} \)
\( \displaystyle +\frac{\partial V_z}{\partial y}W_x \) \( \displaystyle + V_z\frac{\partial W_x}{\partial y} \) \( \displaystyle -\frac{\partial V_x}{\partial y}W_z \) \( \displaystyle -V_x\frac{\partial W_z}{\partial y} \)
\( \displaystyle +\frac{\partial V_x}{\partial z}W_y \) \( \displaystyle + V_x\frac{\partial W_y}{\partial z} \) \( \displaystyle -\frac{\partial V_y}{\partial z}W_x \) \( \displaystyle -V_y\frac{\partial W_x}{\partial z} \)
\( \displaystyle =W_x(\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z})+ W_y(\frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x}) \) \( \displaystyle + W_z(\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y}) \)
\( \displaystyle -V_x(\frac{\partial W_z}{\partial y}-\frac{\partial W_y}{\partial z})+ V_y(\frac{\partial W_x}{\partial z}-\frac{\partial W_z}{\partial x}) \) \( \displaystyle + V_z(\frac{\partial W_y}{\partial x}-\frac{\partial W_x}{\partial y}) \)
\( \displaystyle =\vec{W}\cdot{\rm rot}\vec{V} \) \( \displaystyle -\vec{V}\cdot{\rm rot}\vec{W} \)

\( \displaystyle {\rm div}\hspace{4px}{\rm rot}\vec{V}=0 \)
\( \displaystyle \nabla\cdot(\nabla\times\vec{V})=0 \)

≪定義通りに計算すれば≫
\( \displaystyle {\rm rot}\vec{V}=\begin{vmatrix}\vec{e_x}&\vec{e_y}&\vec{e_z}\\ \dfrac{\partial}{\partial x}& \dfrac{\partial}{\partial y}& \dfrac{\partial}{\partial z}\\ V_x& V_y& V_z\end{vmatrix} \)
\( \displaystyle =\Big(\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z},\hspace{2px} \) \( \displaystyle \frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x},\hspace{3px}\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y}\Big) \)
\( \displaystyle {\rm div}\hspace{4px}{\rm rot}\vec{V} \)
\( \displaystyle =\frac{\partial}{\partial x}(\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z}) \) \( \displaystyle +\frac{\partial}{\partial y}(\frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x})+\frac{\partial}{\partial z}(\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y}) \)
\( \displaystyle =\frac{\partial^2 V_z}{\partial x\partial y}-\frac{\partial^2 V_y}{\partial x\partial z} \) \( \displaystyle +\frac{\partial^2 V_x}{\partial y\partial z}-\frac{\partial^2 V_z}{\partial y\partial x}+\frac{\partial^2 V_y}{\partial z\partial x}-\frac{\partial^2 V_x}{\partial z\partial y} \)
連続関数については，偏微分の順序を交換しても等しいから
\( \displaystyle \frac{\partial^2V_z}{\partial x\partial y}=\frac{\partial^2V_z}{\partial y\partial x} \)
などが成り立つ
したがって
\( \displaystyle {\rm div}\hspace{4px}{\rm rot}\vec{V}=0 \)
≪結果をベクトルの内積・外積で表せば≫
\( \displaystyle {\rm div}\rightarrow\nabla\cdot,\hspace{10px}{\rm rot}\rightarrow\nabla\times \)のように対応させると
\( \displaystyle \nabla\cdot(\nabla\times\vec{V})=0 \)

\( \displaystyle {\rm rot}(\vec{V}+\vec{W})={\rm rot}\vec{V}+{\rm rot}\vec{W} \)
\( \displaystyle \nabla\times(\vec{V}+\vec{W})=\nabla\times\vec{V}+\nabla\times\vec{W} \)

≪定義通りに計算すれば≫
\( \displaystyle \vec{V}+\vec{W}=(V_x,V_y,V_z)+(W_x,W_y,W_z) \)
\( \displaystyle =(V_x+ W_x,V_y+ W_y,V_z+ W_z) \)
\( \displaystyle {\rm rot}(\vec{V}+\vec{W}) \)
\( \displaystyle =\begin{vmatrix}\vec{e_x}&\vec{e_y}&\vec{e_z}\\ \dfrac{\partial}{\partial x}& \dfrac{\partial}{\partial y}& \dfrac{\partial}{\partial z}\\ V_x+ W_x& V_y+ W_y& V_z+ W_z\end{vmatrix} \)
\( \displaystyle =\vec{e_x}\Big\{\frac{\partial}{\partial y}(V_z+ W_z)-\frac{\partial}{\partial z}(V_y+ W_y)\Big\} \)
\( \displaystyle +\vec{e_y}\Big\{\frac{\partial}{\partial z}(V_x+ W_x)-\frac{\partial}{\partial x}(V_z+ W_z)\Big\} \)
\( \displaystyle +\vec{e_z}\Big\{\frac{\partial}{\partial x}(V_y+ W_y)-\frac{\partial}{\partial y}(V_x+ W_x)\Big\} \)
\( \displaystyle =\vec{e_x}\Big\{(\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z})+(\frac{\partial W_z}{\partial y}-\frac{\partial W_y}{\partial z})\Big\} \)
\( \displaystyle +\vec{e_y}\Big\{(\frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x})+(\frac{\partial W_x}{\partial z}-\frac{\partial W_z}{\partial x})\Big\} \)
\( \displaystyle +\vec{e_z}\Big\{(\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y})+(\frac{\partial W_y}{\partial x}-\frac{\partial W_x}{\partial y})\Big\} \)
\( \displaystyle =\Big(\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z},\hspace{5px}\frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x} \) \( \displaystyle ,\hspace{5px}\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y}\Big) \)
\( \displaystyle +\Big(\frac{\partial W_z}{\partial y}-\frac{\partial W_y}{\partial z},\hspace{5px}\frac{\partial W_x}{\partial z}-\frac{\partial W_z}{\partial x} \) \( \displaystyle ,\hspace{5px}\frac{\partial W_y}{\partial x}-\frac{\partial W_x}{\partial y}\Big) \)
\( \displaystyle ={\rm rot}\vec{V}+{\rm rot}\vec{W} \)
≪ベクトルの内積・外積で表せば≫
\( \displaystyle \nabla\times(\vec{V}+\vec{W})=\nabla\times\vec{V}+\nabla\times\vec{W} \)

\( \displaystyle {\rm rot}(\phi\vec{V})=({\rm grad}\hspace{2px}\phi)\times\vec{V}+\phi\hspace{2px}{\rm rot}\vec{V} \)
\( \displaystyle \nabla\times(\phi\vec{V})=(\nabla\phi)\times\vec{V}+\phi(\nabla\times\vec{V}) \)

≪定義通りに計算すれば≫
\( \displaystyle \phi\vec{V}=(\phi V_x,\hspace{4px}\phi V_y,\hspace{4px}\phi V_z) \)
\( \displaystyle {\rm rot}(\phi\vec{V})=\begin{vmatrix}\vec{e_x}&\vec{e_y}&\vec{e_z}\\ \dfrac{\partial}{\partial x}& \dfrac{\partial}{\partial y}& \dfrac{\partial}{\partial z}\\ \phi V_x& \phi V_y& \phi V_z\end{vmatrix} \)
\( \displaystyle =\vec{e_x}\Big\{\frac{\partial}{\partial y}(\phi V_z)-\frac{\partial}{\partial z}(\phi V_y)\Big\} \)
\( \displaystyle +\vec{e_y}\Big\{\frac{\partial}{\partial z}(\phi V_x)-\frac{\partial}{\partial x}(\phi V_z)\Big\} \)
\( \displaystyle +\vec{e_z}\Big\{\frac{\partial}{\partial x}(\phi V_y)-\frac{\partial}{\partial y}(\phi V_x)\Big\} \)
積の微分法：\( \displaystyle \frac{\partial}{\partial y}(\phi V_z)=\frac{\partial \phi}{\partial y}V_z+ \phi\frac{\partial V_z}{\partial y} \)
などにより
\( \displaystyle \vec{e_x}\Big\{\frac{\partial\phi}{\partial y}V_z+\phi\frac{\partial V_z}{\partial y}-\frac{\partial\phi}{\partial z}V_y-\phi\frac{\partial V_y}{\partial z}\Big\} \)
\( \displaystyle +\vec{e_y}\Big\{\frac{\partial\phi}{\partial z}V_x+\phi\frac{\partial V_x}{\partial z}-\frac{\partial\phi}{\partial x}V_z-\phi\frac{\partial V_z}{\partial x}\Big\} \)
\( \displaystyle +\vec{e_z}\Big\{\frac{\partial\phi}{\partial x}V_y+\phi\frac{\partial V_y}{\partial x}-\frac{\partial\phi}{\partial y}V_x-\phi\frac{\partial V_x}{\partial y}\Big\} \)
\( \displaystyle =\vec{e_x}\Big\{\frac{\partial\phi}{\partial y}V_z-\frac{\partial\phi}{\partial z}V_y+\phi(\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z})\Big\} \)
\( \displaystyle +\vec{e_y}\Big\{\frac{\partial\phi}{\partial z}V_x-\frac{\partial\phi}{\partial x}V_z+\phi(\frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x})\Big\} \)
\( \displaystyle +\vec{e_z}\Big\{\frac{\partial\phi}{\partial x}V_y-\frac{\partial\phi}{\partial y}V_x+\phi(\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y})\Big\} \)
\( \displaystyle =\begin{vmatrix}\vec{e_x}&\vec{e_y}&\vec{e_z}\\ \dfrac{\partial\phi}{\partial x}& \dfrac{\partial\phi}{\partial y}& \dfrac{\partial\phi}{\partial z}\\ V_x& V_y& V_z\end{vmatrix} \) \( \displaystyle +\phi\begin{vmatrix}\vec{e_x}&\vec{e_y}&\vec{e_z}\\ \dfrac{\partial}{\partial x}& \dfrac{\partial}{\partial y}& \dfrac{\partial}{\partial z}\\ V_x& V_y& V_z\end{vmatrix} \)
\( \displaystyle =({\rm grad}\hspace{2px}\phi)\times\vec{V}+\phi\hspace{2px}{\rm rot}\vec{V} \)
≪ベクトルの内積・外積で表せば≫
\( \displaystyle {\rm grad}\rightarrow\nabla,\hspace{10px}{\rm rot}\rightarrow\nabla\times \)と対応させると
\( \displaystyle \nabla\times(\phi\vec{V})=(\nabla\phi)\times\vec{V}+\phi(\nabla\times\vec{V}) \)

\( \displaystyle {\rm rot}\hspace{4px}{\rm grad}\hspace{4px}\phi=\vec{0} \)
\( \displaystyle \nabla\times(\nabla\phi)=\vec{0} \)

≪定義通りに計算すれば≫
\( \displaystyle {\rm grad}\phi=(\frac{\partial\phi}{\partial x},\hspace{2px}\frac{\partial\phi}{\partial y},\hspace{2px}\frac{\partial\phi}{\partial z}) \)
\( \displaystyle {\rm rot}\hspace{5px}{\rm grad}\hspace{5px}\phi \)
\( \displaystyle =\vec{e_x}\Big\{\frac{\partial}{\partial y}(\frac{\partial\phi}{\partial z})-\frac{\partial}{\partial z}(\frac{\partial\phi}{\partial y})\Big\} \)
\( \displaystyle +\vec{e_y}\Big\{\frac{\partial}{\partial z}(\frac{\partial\phi}{\partial x})-\frac{\partial}{\partial x}(\frac{\partial\phi}{\partial z})\Big\} \)
\( \displaystyle +\vec{e_z}\Big\{\frac{\partial}{\partial x}(\frac{\partial\phi}{\partial y})-\frac{\partial}{\partial y}(\frac{\partial\phi}{\partial x})\Big\} \)
連続関数については，偏微分の順序を交換しても等しいから
\( \displaystyle \frac{\partial}{\partial y}(\frac{\partial\phi}{\partial z})=\frac{\partial}{\partial z}(\frac{\partial\phi}{\partial y}) \)
などが成り立つ
したがって，\( \displaystyle {\rm rot}\hspace{5px}{\rm grad}\hspace{5px}\phi=\vec{0} \)
≪ベクトルの内積・外積で表せば≫
\( \displaystyle {\rm rot}\rightarrow\nabla\times,\hspace{10px}{\rm grad}\rightarrow\nabla \)のように対応させると
\( \displaystyle \nabla\times(\nabla\phi)=\vec{0} \)

\( \displaystyle {\rm rot}\hspace{4px}{\rm rot}\hspace{4px}\vec{V}={\rm grad}\hspace{4px}{\rm div}\hspace{4px}\vec{V}-\Delta\vec{V} \)
\( \displaystyle \nabla\times(\nabla\times\vec{V})=\nabla(\nabla\cdot\vec{V})-\Delta\vec{V} \)

≪定義通りに計算すれば≫
\( \displaystyle {\rm rot}\vec{V}=\begin{vmatrix}\vec{e_x}&\vec{e_y}&\vec{e_z}\\ \dfrac{\partial}{\partial x}& \dfrac{\partial}{\partial y}& \dfrac{\partial}{\partial z}\\ V_x& V_y& V_z\end{vmatrix} \)
\( \displaystyle =\Big(\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z},\hspace{3px} \) \( \displaystyle \frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x},\hspace{3px}\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y}\Big) \)
\( \displaystyle {\rm rot}\hspace{4px}{\rm rot}\vec{V} \)
\( \displaystyle =\vec{e_x}\Big\{\frac{\partial}{\partial y}(\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y}) \) \( \displaystyle -\frac{\partial}{\partial z}(\frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x})\Big\} \)
\( \displaystyle +\vec{e_y}\Big\{\frac{\partial}{\partial z}(\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y}) \) \( \displaystyle -\frac{\partial}{\partial x}(\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z})\Big\} \)
\( \displaystyle +\vec{e_z}\{\frac{\partial}{\partial x}(\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z}) \) \( \displaystyle -\frac{\partial}{\partial y}(\frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x})\Big\} \)
x成分は
\( \displaystyle \frac{\partial^2}{\partial y\partial x}V_y-\frac{\partial^2}{\partial y^2}V_x-\frac{\partial^2}{\partial z^2}V_x+\frac{\partial^2}{\partial z\partial x}V_z \)
だから，\( \displaystyle \frac{\partial^2}{\partial x^2}V_x \)を足して引くと
\( \displaystyle \frac{\partial^2}{\partial x^2}V_x+\frac{\partial^2}{\partial y\partial x}V_y+\frac{\partial^2}{\partial z\partial x}V_z \)\( \displaystyle -(\frac{\partial^2}{\partial x^2}V_x+\frac{\partial^2}{\partial y^2}V_x+\frac{\partial^2}{\partial z^2}V_x) \)
\( \displaystyle =\frac{\partial}{\partial x}(\frac{\partial V_x}{\partial x}+\frac{\partial V_y}{\partial y}+\frac{\partial V_z}{\partial z}) \) \( \displaystyle -(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})V_x \)
\( \displaystyle =\frac{\partial}{\partial x}\rm{div}\vec{V}-\Delta V_x \)
同様にして，ｙ，ｚ成分は各々，次のようになる
\( \displaystyle \frac{\partial}{\partial y}\rm{div}\vec{V}-\Delta V_y \)
\( \displaystyle \frac{\partial}{\partial x}\rm{div}\vec{V}-\Delta V_z \)
まとめると
\( \displaystyle (\frac{\partial}{\partial x},\hspace{3px}\frac{\partial}{\partial y},\hspace{3px}\frac{\partial}{\partial z}){\rm div}\vec{V}-\Delta(V_x,\hspace{3px}V_y,\hspace{3px}V_z) \)
\( \displaystyle ={\rm grad}\hspace{4px}{\rm div}\hspace{4px}\vec{V}-\Delta\vec{V} \)
≪結果をベクトルの内積・外積で表せば≫
\( \displaystyle {\rm rot}\rightarrow\nabla\times,\hspace{10px}{\rm grad}\rightarrow\nabla,\hspace{10px}{\rm div}\rightarrow\nabla\cdot \)のように対応させると
\( \displaystyle {\rm rot}\hspace{4px}{\rm rot}\hspace{4px}\vec{V}={\rm grad}\hspace{4px}{\rm div}\hspace{4px}\vec{V}-\Delta\vec{V} \)
は，次の形に書ける
\( \displaystyle \nabla\times(\nabla\times\vec{V})=\nabla(\nabla\cdot\vec{V})-\Delta\vec{V} \)

\( \displaystyle \Delta(\phi\psi)=\psi\Delta\phi+\phi\Delta\psi+ 2{\rm grad}\phi\cdot{\rm grad}\psi \)
\( \displaystyle \Delta(\phi\psi)=\psi\Delta\phi+\phi\Delta\psi+ 2\nabla\phi\cdot\nabla\psi \)

≪定義通りに計算すれば≫
\( \displaystyle \Delta(\phi\psi)=\frac{\partial^2(\phi\psi)}{\partial x^2}+\frac{\partial^2(\phi\psi)}{\partial y^2}+\frac{\partial^2(\phi\psi)}{\partial z^2} \)
xによる微分の部分は
\( \displaystyle \frac{\partial (\phi\psi)}{\partial x}=\frac{\partial \phi}{\partial x}\psi+\phi\frac{\partial \psi}{\partial x} \)
\( \displaystyle \frac{\partial}{\partial x}\Big(\frac{\partial (\phi\psi)}{\partial x}\Big) \) \( \displaystyle =\frac{\partial}{\partial x}\Big(\frac{\partial \phi}{\partial x}\psi\Big)+\frac{\partial}{\partial x}\Big(\phi\frac{\partial \psi}{\partial x}\Big) \)
\( \displaystyle =\frac{\partial^2\phi}{\partial x^2}\psi+\frac{\partial\phi}{\partial x}\frac{\partial \psi}{\partial x} \) \( \displaystyle +\frac{\partial\phi}{\partial x}\frac{\partial \psi}{\partial x}+\phi\frac{\partial^2\psi}{\partial x^2} \)
\( \displaystyle =\frac{\partial^2\phi}{\partial x^2}\psi+ 2\frac{\partial\phi}{\partial x}\frac{\partial \psi}{\partial x}+\phi\frac{\partial^2\psi}{\partial x^2} \)
同様にして，y,zによる微分の部分は
\( \displaystyle \frac{\partial^2\phi}{\partial y^2}\psi+ 2\frac{\partial\phi}{\partial y}\frac{\partial \psi}{\partial y}+\phi\frac{\partial^2\psi}{\partial y^2} \)
\( \displaystyle \frac{\partial^2\phi}{\partial z^2}\psi+ 2\frac{\partial\phi}{\partial z}\frac{\partial \psi}{\partial z}+\phi\frac{\partial^2\psi}{\partial z^2} \)
これらを加えると
\( \displaystyle \frac{\partial^2\phi}{\partial x^2}\psi+\frac{\partial^2\phi}{\partial y^2}\psi+\frac{\partial^2\phi}{\partial z^2}\psi \)
\( \displaystyle + 2(\frac{\partial\phi}{\partial x}\frac{\partial \psi}{\partial x}+\frac{\partial\phi}{\partial y}\frac{\partial \psi}{\partial y}+\frac{\partial\phi}{\partial z}\frac{\partial \psi}{\partial z}) \)
\( \displaystyle +\phi\frac{\partial^2\psi}{\partial x^2}\phi\frac{\partial^2\psi}{\partial y^2}\phi\frac{\partial^2\psi}{\partial z^2} \)
\( \displaystyle =\psi\Delta\phi+\phi\Delta\psi+ 2{\rm grad}\phi\cdot{\rm grad}\psi \)