ベクトル解析 k-san.link

ベクトル解析の公式と証明

ベクトル解析の諸公式と，その証明を示します．ナブラやラプラシアン，ベクトル解析の計算に関する基本事項についても，簡単に述べます．

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ベクトル解析の公式

１．次の公式が成り立つ．
（１）

(1)   \begin{equation*} \frac{d}{dt} (\phi {\bf A}) = \phi \frac{d {\bf A}}{dt} + \frac{d \phi }{dt} {\bf A} \end{equation*}

【 式（１）の証明を見る 】

式（１）の証明

    \begin{eqnarray*} &&\frac{d}{dt} (\phi {\bf A})\\ &=& \frac{d}{dt} \left( \phi A_x, \phi A_y, \phi A_z \right) \\ &=& \left( \frac{d(\phi A_x)}{dt}, \frac{d(\phi A_y)}{dt}, \frac{d(\phi A_z)}{dt} \right) \\ &=& \left( \phi \frac{ d A_x}{dt} + \frac{d\phi}{dt}A_x, \phi \frac{ d A_y}{dt} + \frac{d\phi}{dt}A_y, \phi \frac{ d A_z}{dt} + \frac{d\phi}{dt}A_z \right) \\ &=& \left( \phi \frac{ d A_x}{dt}, \phi \frac{ d A_y}{dt}, \phi \frac{ d A_z}{dt} \right) + \left( \frac{d\phi}{dt}A_x, \frac{d\phi}{dt}A_y, \frac{d\phi}{dt}A_z \right) \\ &=& \phi \left( \frac{ d A_x}{dt}, \frac{ d A_y}{dt}, \frac{ d A_z}{dt} \right) + \frac{d\phi}{dt} \left( A_x, A_y, A_z \right) \\ &=& \phi \frac{d {\bf A}}{dt} + \frac{d \phi }{dt} {\bf A} \end{eqnarray*}

 
（２）

(2)   \begin{equation*} \frac{d}{dt} ({\bf A} \cdot {\bf B}) = {\bf A} \cdot \frac{d {\bf B}}{dt} + \frac{d {\bf A} }{dt} \cdot {\bf B} \end{equation*}

【 式（２）の証明を見る 】

式（２）の証明

    \begin{eqnarray*} &&\frac{d}{dt} ({\bf A} \cdot {\bf B})\\ &=& \frac{d}{dt} \left( A_x　B_x + A_y B_y + A_z B_z \right) \\ &=& \frac{d( A_x B_x)}{dt} + \frac{d( A_y B_y)}{dt} + \frac{d( A_z B_z )}{dt} \\ &=& A_x \frac{ d B_x}{dt} + \frac{d A_x}{dt}B_x + A_y \frac{ d B_y}{dt} + \frac{d A_y }{dt}B_y + A_z \frac{ d B_z}{dt} + \frac{d A_z}{dt}B_z \\ &=& \left( A_x \frac{ d B_x}{dt} + A_y \frac{ d B_y}{dt} + A_z \frac{ d B_z}{dt} \right) \\ && \qquad + \left( \frac{d A_x}{dt}B_x + \frac{d A_y }{dt}B_y + \frac{d A_z}{dt}B_z \right) \\ &=& \left( A_x, A_y, A_z \right) \cdot \left( \frac{ d B_x}{dt}, \frac{ d B_y}{dt}, \frac{ d B_z}{dt} \right) \\ && \qquad + \left( \frac{d A_x}{dt}, \frac{d A_y }{dt}, \frac{d A_z}{dt} \right) \cdot \left( B_x, B_y, B_z \right) \\ &=& {\bf A} \cdot \frac{d {\bf B}}{dt} + \frac{d {\bf A} }{dt} \cdot {\bf B} \end{eqnarray*}

 
（３）

(3)   \begin{equation*} \frac{d}{dt} ({\bf A} \times {\bf B}) = {\bf A} \times \frac{d {\bf B}}{dt} + \frac{d {\bf A} }{dt} \times {\bf B} \end{equation*}

【 式（３）の証明を見る 】

式（３）の証明
式(3)左辺：

    \begin{eqnarray*} && \frac{d}{dt} ({\bf A} \times {\bf B}) \\ &=& \frac{d}{dt} \left| \begin{array}{ccc} {\bf e}_x & {\bf e}_y & {\bf e}_z\\ A_x & A_y & A_z\\ B_x & B_y & B_z \end{array} \right| \\ &=& \frac{d}{dt} \left( {\bf e}_x \left| \begin{array}{cc} A_y & A_z\\ B_y & B_z \end{array} \right| - {\bf e}_y \left| \begin{array}{cc} A_x & A_z\\ B_x & B_z \end{array} \right| + {\bf e}_z \left| \begin{array}{cc} A_x & A_y \\ B_x & B_y \end{array} \right| \right) \\ &=& \frac{d}{dt} \bigg( {\bf e}_x \left( A_y　B_z - A_z B_y \right) +{\bf e}_y \left( A_z　B_x - A_x B_z \right) +{\bf e}_z \left( A_x　B_y - A_y B_x \right) \bigg)\\ &=& {\bf e}_x \left( A_y　\frac{dB_z}{dt} + \frac{dA_y}{dt}　B_z - A_z \frac{dB_y}{dt} - \frac{dA_z}{dt} B_y \right)\\ &&\quad +{\bf e}_y \left( A_z　\frac{dB_x}{dt} + \frac{dA_z}{dt}　B_x - A_x \frac{dB_z}{dt} - \frac{dA_x}{dt} B_z \right)\\ &&\qquad +{\bf e}_z \left( A_x　\frac{dB_y}{dt} + \frac{dA_x}{dt}　B_y - A_y \frac{dB_x}{dt} - \frac{dA_y}{dt} B_x \right) \end{eqnarray*}

式(3)右辺：

    \begin{eqnarray*} && {\bf A} \times \frac{d {\bf B}}{dt} + \frac{d {\bf A} }{dt} \times {\bf B} \\ &=& \left| \begin{array}{ccc} {\bf e}_x & {\bf e}_y & {\bf e}_z\\ A_x & A_y & A_z\\ \frac{dB_x}{dt} & \frac{dB_y}{dt} & \frac{dB_z}{dt} \end{array} \right| \\ && + \left| \begin{array}{ccc} {\bf e}_x & {\bf e}_y & {\bf e}_z\\ \frac{dA_x}{dt} & \frac{dA_y}{dt} & \frac{dA_z}{dt} \\ B_x & B_y & B_z \end{array} \right| \\ &=& \left( {\bf e}_x \left| \begin{array}{cc} A_y & A_z\\ \frac{dB_y}{dt} & \frac{dB_z}{dt} \end{array} \right| - {\bf e}_y \left| \begin{array}{cc} A_x & A_z\\ \frac{dB_x}{dt} & \frac{dB_z}{dt} \end{array} \right| + {\bf e}_z \left| \begin{array}{cc} A_x & A_y \\ \frac{dB_x}{dt} & \frac{dB_y}{dt} \end{array} \right| \right)\\ &&+ \left( {\bf e}_x \left| \begin{array}{cc} \frac{dA_y}{dt} & \frac{dA_z}{dt} \\ B_y & B_z \end{array} \right| - {\bf e}_y \left| \begin{array}{cc} \frac{dA_x}{dt} & \frac{dA_z}{dt} \\ B_x & B_z \end{array} \right| + {\bf e}_z \left| \begin{array}{cc} \frac{dA_x}{dt} & \frac{dA_y}{dt} \\ B_x & B_y \end{array} \right| \right)\\ &=& {\bf e}_x \left( A_y　\frac{dB_z}{dt} + \frac{dA_y}{dt}　B_z - A_z \frac{dB_y}{dt} - \frac{dA_z}{dt} B_y \right)\\ &&\quad +{\bf e}_y \left( A_z　\frac{dB_x}{dt} + \frac{dA_z}{dt}　B_x - A_x \frac{dB_z}{dt} - \frac{dA_x}{dt} B_z \right)\\ &&\qquad +{\bf e}_z \left( A_x　\frac{dB_y}{dt} + \frac{dA_x}{dt}　B_y - A_y \frac{dB_x}{dt} - \frac{dA_y}{dt} B_x \right) \end{eqnarray*}

よって，左辺と右辺は等しいため，式(3)は成り立つ．

 

２．次の公式が成り立つ．
（４）

(4)   \begin{equation*} \nabla (\phi \psi) = \phi (\nabla \psi )+ \psi (\nabla \phi) \end{equation*}

【 式（４）の証明を見る 】

式（４）の証明

    \begin{eqnarray*} && \nabla (\phi \psi) \\ &=& \left( \frac{\partial }{\partial x }, \frac{\partial }{\partial y }, \frac{\partial}{\partial z } \right)(\phi \psi)\\ &=& \left( {\bf e}_x \frac{\partial }{\partial x }+{\bf e}_y\frac{\partial }{\partial y }+{\bf e}_z\frac{\partial}{\partial z } \right)(\phi \psi)\\ &=& {\bf e}_x \frac{\partial(\phi \psi) }{\partial x }+{\bf e}_y\frac{\partial(\phi \psi) }{\partial y }+{\bf e}_z\frac{\partial(\phi \psi) }{\partial z }\\ &=& {\bf e}_x\left( \phi \frac{\partial \psi }{\partial x }+\psi \frac{\partial \phi }{\partial x } \right)\\ && \quad +{\bf e}_y\left( \phi \frac{\partial \psi }{\partial y }+\psi \frac{\partial \phi }{\partial y } \right)\\ && \qquad +{\bf e}_z\left( \phi \frac{\partial \psi }{\partial z }+\psi \frac{\partial \phi }{\partial z } \right)\\ &=& \left( {\bf e}_x \phi \frac{\partial \psi }{\partial x } + {\bf e}_y \phi \frac{\partial \psi }{\partial y } + {\bf e}_z \phi \frac{\partial \psi }{\partial z }\right) \\ && \quad + \left( {\bf e}_x \psi \frac{\partial \phi }{\partial x } + {\bf e}_y \psi \frac{\partial \phi }{\partial y } +{\bf e}_z \psi \frac{\partial \phi }{\partial z } \right)\\ &=& \phi \left( \frac{\partial \psi }{\partial x } , \frac{\partial \psi }{\partial y }, \frac{\partial \psi }{\partial z }\right) + \psi \left( \frac{\partial \phi }{\partial x } , \frac{\partial \phi }{\partial y } , \frac{\partial \phi }{\partial z } \right)\\ &=& \phi \left[ \left( \frac{\partial }{\partial x } , \frac{\partial }{\partial y }, \frac{\partial }{\partial z }\right) \psi \right] + \psi \left[ \left( \frac{\partial }{\partial x } , \frac{\partial }{\partial y } , \frac{\partial }{\partial z } \right)\phi \right]\\ &=& \phi (\nabla \psi )+ \psi (\nabla \phi) \end{eqnarray*}

 
（５）

(5)   \begin{equation*} \nabla \cdot (\phi {\bf A}) = \phi (\nabla \cdot {\bf A}) + (\nabla \phi) \cdot {\bf A} \end{equation*}

【 式（５）の証明を見る 】

式（５）の証明

    \begin{eqnarray*} &&\nabla \cdot (\phi {\bf A}) \\ &=& \left( \frac{\partial }{\partial x }, \frac{\partial }{\partial y }, \frac{\partial}{\partial z } \right)\cdot (\phi {\bf A})\\ &=& \left( \frac{\partial }{\partial x }, \frac{\partial }{\partial y }, \frac{\partial}{\partial z } \right)\cdot \left( \phi A_x , \phi A_y , \phi A_z \right)\\ &=& \frac{\partial (\phi A_x)}{\partial x } + \frac{\partial (\phi A_y)}{\partial y } + \frac{\partial (\phi A_z)}{\partial z } \\ &=& \phi \frac{\partial A_x}{\partial x } + \frac{\partial \phi }{\partial x } A_x \\ && \quad + \phi \frac{\partial A_y}{\partial y } + \frac{\partial \phi }{\partial y } A_y \\ && \qquad + \phi \frac{\partial A_z}{\partial z } + \frac{\partial \phi }{\partial z } A_z \\ &=& \left( \phi \frac{\partial A_x}{\partial x } + \phi \frac{\partial A_y}{\partial y } + \phi \frac{\partial A_z}{\partial z } \right) \\ && \quad + \left( \frac{\partial \phi }{\partial x } A_x + \frac{\partial \phi }{\partial y } A_y + \frac{\partial \phi }{\partial z } A_z \right) \\ &=& \phi \left( \frac{\partial A_x}{\partial x } + \frac{\partial A_y}{\partial y } + \frac{\partial A_z}{\partial z } \right) \\ && \quad + \left( \frac{\partial \phi }{\partial x }, \frac{\partial \phi }{\partial y }, \frac{\partial \phi }{\partial z } \right) \cdot \left( A_x, A_y, A_z \right)\\ &=& \phi \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \cdot \left( A_x, A_y, A_z \right) \\ && \quad + \left[ \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right)\phi \right] \cdot \left( A_x, A_y, A_z \right)\\ &=& \phi (\nabla \cdot {\bf A}) + (\nabla \phi) \cdot {\bf A} \end{eqnarray*}

 

（６）

(6)   \begin{equation*} \nabla \times (\phi {\bf A}) = \phi (\nabla \times {\bf A}) + (\nabla \phi) \times {\bf A} \end{equation*}

【 式（６）の証明を見る 】

式（６）の証明

    \begin{eqnarray*} &&\nabla \times (\phi {\bf A}) \\ &=& \left( \frac{\partial }{\partial x }, \frac{\partial }{\partial y }, \frac{\partial}{\partial z } \right)\times (\phi {\bf A})\\ &=& \left( \frac{\partial }{\partial x }, \frac{\partial }{\partial y }, \frac{\partial}{\partial z } \right) \times \left( \phi A_x , \phi A_y , \phi A_z \right)\\ &=& \left| \begin{array}{ccc} {\bf e}_x & {\bf e}_y & {\bf e}_z\\ \frac{\partial }{\partial x }& \frac{\partial }{\partial y }& \frac{\partial}{\partial z }\\ \phi A_x & \phi A_y & \phi A_z \end{array} \right| \\ &=& {\bf e}_x \left| \begin{array}{cc} \frac{\partial }{\partial y }& \frac{\partial}{\partial z }\\ \phi A_y & \phi A_z \end{array} \right| - {\bf e}_y \left| \begin{array}{cc} \frac{\partial }{\partial x }& \frac{\partial}{\partial z }\\ \phi A_x & \phi A_z \end{array} \right| + {\bf e}_z \left| \begin{array}{cc} \frac{\partial }{\partial x }& \frac{\partial}{\partial y } \\ \phi A_x & \phi A_y \end{array} \right| \\ &=& {\bf e}_x \left( \frac{\partial (\phi A_z)}{\partial y } - \frac{\partial (\phi A_y)}{\partial z } \right) \\ && \quad + {\bf e}_y \left( \frac{\partial (\phi A_x)}{\partial z } - \frac{\partial (\phi A_z)}{\partial x } \right) \\ && \qquad + {\bf e}_z \left( \frac{\partial (\phi A_y)}{\partial x } - \frac{\partial (\phi A_x)}{\partial y } \right) \\ &=& {\bf e}_x \left( \phi \frac{\partial A_z}{\partial y } + \frac{\partial \phi}{\partial y }A_z - \phi \frac{\partial A_y}{\partial z } - \frac{\partial \phi}{\partial z }A_y \right) \\ && \quad + {\bf e}_y \left( \phi \frac{\partial A_x}{\partial z } + \frac{\partial \phi}{\partial z }A_x - \phi \frac{\partial A_z}{\partial x } - \frac{\partial \phi}{\partial x }A_z \right) \\ &&\qquad + {\bf e}_z \left( \phi \frac{\partial A_y}{\partial x } + \frac{\partial \phi}{\partial x }A_y - \phi \frac{\partial A_x}{\partial y } - \frac{\partial \phi}{\partial y }A_x\right) \\ &=& {\bf e}_x \left( \phi\frac{\partial A_z}{\partial y} -\phi \frac{\partial A_y}{\partial z}\right) + {\bf e}_x \left(\frac{\partial \phi}{\partial y}A_z -\frac{\partial \phi}{\partial z }A_y \right) \\ && \quad + {\bf e}_y \left( \phi\frac{\partial A_x}{\partial z} -\phi \frac{\partial A_z}{\partial x}\right) + {\bf e}_y \left(\frac{\partial \phi}{\partial z}A_x -\frac{\partial \phi}{\partial x }A_z \right) \\ &&\qquad + {\bf e}_z \left( \phi\frac{\partial A_y}{\partial x} -\phi \frac{\partial A_x}{\partial y}\right) + {\bf e}_z \left(\frac{\partial \phi}{\partial x}A_y -\frac{\partial \phi}{\partial y }A_x\right) \\ &=& \phi \left [ {\bf e}_x \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) + {\bf e}_y \left( \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) + {\bf e}_z \left( \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) \right]\\ && + {\bf e}_x \left(\frac{\partial \phi}{\partial y}A_z -\frac{\partial \phi}{\partial z }A_y \right) + {\bf e}_y \left(\frac{\partial \phi}{\partial z}A_x -\frac{\partial \phi}{\partial x }A_z \right) + {\bf e}_z \left(\frac{\partial \phi}{\partial x}A_y -\frac{\partial \phi}{\partial y }A_x\right) \\ &=& \phi \left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} , \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x} , \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \right)\\ && \quad + \left(\frac{\partial \phi}{\partial y}A_z -\frac{\partial \phi}{\partial z }A_y , \frac{\partial \phi}{\partial z}A_x -\frac{\partial \phi}{\partial x }A_z , \frac{\partial \phi}{\partial x}A_y -\frac{\partial \phi}{\partial y }A_x \right) \\ &=& \phi \left[ \left( \frac{\partial }{\partial x }, \frac{\partial }{\partial y }, \frac{\partial}{\partial z } \right) \times \left( A_x , A_y , A_z \right) \right] \\ && \quad + \left( \frac{\partial \phi}{\partial x }, \frac{\partial \phi}{\partial y }, \frac{\partial \phi}{\partial z } \right) \times \left( A_x , A_y , A_z \right) \\ &=& \phi (\nabla \times {\bf A}) + (\nabla \phi) \times {\bf A} \end{eqnarray*}

 

（７）

(7)   \begin{eqnarray*} \nabla \cdot ({\bf A} \times {\bf B}) &=& (\nabla \times {\bf A}) \cdot {\bf B} + {\bf A} \cdot ({\bf B} \times \nabla) \\ &=& (\nabla \times {\bf A}) \cdot {\bf B} - {\bf A} \cdot (\nabla \times {\bf B} ) \end{eqnarray*}

【 式（７）の証明を見る 】

式（７）の証明

    \begin{eqnarray*} && \nabla \cdot ({\bf A} \times {\bf B}) \\ &=& \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \cdot \Big( \left( A_x , A_y , A_z \right) \times \left( B_x , B_y , B_z \right) \Big) \\ &=& \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \cdot \Big( A_y　B_z - A_z B_y, \; A_z　B_x - A_x B_z, \; A_x　B_y - A_y B_x \Big)\\ &=& \frac{\partial}{\partial x }( A_y　B_z - A_z B_y) + \frac{\partial}{\partial y}(A_z　B_x - A_x B_z)+ \frac{\partial}{\partial z } ( A_x　B_y - A_y B_x )\\ &=& \left( A_y　\frac{\partial B_z}{\partial x } + \frac{\partial A_y}{\partial x }　B_z - A_z \frac{\partial B_y}{\partial x } - \frac{\partial A_z}{\partial x } B_y \right) \\ && \quad + \left( A_z　\frac{\partial B_x}{\partial y } + \frac{\partial A_z}{\partial y }　B_x - A_x \frac{\partial B_z}{\partial y } - \frac{\partial A_x}{\partial y } B_z \right) \\ &&\qquad + \left( A_x　\frac{\partial B_y}{\partial z } + \frac{\partial A_x}{\partial z }　B_y - A_y \frac{\partial B_x}{\partial z } - \frac{\partial A_y}{\partial z } B_x \right) \\ &=& \left( A_x　\frac{\partial B_y}{\partial z } - A_x \frac{\partial B_z}{\partial y } \right) + \left( A_y　\frac{\partial B_z}{\partial x } - A_y \frac{\partial B_x}{\partial z } \right) + \left( A_z　\frac{\partial B_x}{\partial y } - A_z \frac{\partial B_y}{\partial x } \right) \\ && + \left( \frac{\partial A_z}{\partial y }　B_x - \frac{\partial A_y}{\partial z } B_x \right) + \left( \frac{\partial A_x}{\partial z }　B_y - \frac{\partial A_z}{\partial x } B_y \right) + \left( \frac{\partial A_y}{\partial x }　B_z - \frac{\partial A_x}{\partial y } B_z \right) \\ &=& A_x \left( \frac{\partial B_y}{\partial z } - \frac{\partial B_z}{\partial y } \right) + A_y \left( \frac{\partial B_z}{\partial x } - \frac{\partial B_x}{\partial z } \right) + A_z \left( \frac{\partial B_x}{\partial y } - \frac{\partial B_y}{\partial x } \right) \\ && + \left( \frac{\partial A_z}{\partial y } - \frac{\partial A_y}{\partial z } \right) B_x + \left( \frac{\partial A_x}{\partial z } - \frac{\partial A_z}{\partial x } \right) B_y + \left( \frac{\partial A_y}{\partial x } - \frac{\partial A_x}{\partial y } \right) B_z\\ &=& \left( A_x , A_y , A_z \right) \cdot \left( \frac{\partial B_y}{\partial z } - \frac{\partial B_z}{\partial y }, \; \frac{\partial B_z}{\partial x } - \frac{\partial B_x}{\partial z }, \; \frac{\partial B_x}{\partial y } - \frac{\partial B_y}{\partial x } \right) \\ && + \left( \frac{\partial A_z}{\partial y } - \frac{\partial A_y}{\partial z }, \; \frac{\partial A_x}{\partial z } - \frac{\partial A_z}{\partial x }, \; \frac{\partial A_y}{\partial x } - \frac{\partial A_x}{\partial y } \right) \cdot \left( B_x , B_y , B_z \right) \\ &=& {\bf A} \cdot ({\bf B} \times \nabla) + (\nabla \times {\bf A}) \cdot {\bf B} \\ &=& (\nabla \times {\bf A}) \cdot {\bf B} + {\bf A} \cdot ({\bf B} \times \nabla) \\ &=& (\nabla \times {\bf A}) \cdot {\bf B} - {\bf A} \cdot (\nabla \times {\bf B} ) \end{eqnarray*}

 

（８）

(8)   \begin{equation*} \nabla \times ({\bf A} \times {\bf B}) = {\bf A} (\nabla \cdot {\bf B}) - ({\bf A} \cdot \nabla) {\bf B} + ({\bf B} \cdot \nabla) {\bf A} - {\bf B} ( \nabla \cdot {\bf A}) \end{equation*}

【 式（８）の証明を見る 】

式（８）の証明

    \begin{eqnarray*} && \nabla \times ({\bf A} \times {\bf B}) \\ &=& \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \times \left[ \left( A_x , A_y , A_z \right) \times \left( B_x , B_y , B_z \right) \right] \\ &=& \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \times \Big( A_y　B_z - A_z B_y, \; A_z　B_x - A_x B_z, \; A_x　B_y - A_y B_x \Big)\\ &=& \left| \begin{array}{ccc} {\bf e}_x & {\bf e}_y & {\bf e}_z\\ \frac{\partial }{\partial x }& \frac{\partial }{\partial y }& \frac{\partial}{\partial z }\\ A_y　B_z - A_z B_y & A_z　B_x - A_x B_z & A_x　B_y - A_y B_x \end{array} \right| \\ &=& {\bf e}_x \left| \begin{array}{cc} \frac{\partial }{\partial y }& \frac{\partial}{\partial z }\\ A_z　B_x - A_x B_z & A_x　B_y - A_y B_x \end{array} \right| \\ && \quad - {\bf e}_y \left| \begin{array}{cc} \frac{\partial }{\partial x }& \frac{\partial}{\partial z }\\ A_y　B_z - A_z B_y & A_x　B_y - A_y B_x \end{array} \right| \\ && \qquad + {\bf e}_z \left| \begin{array}{cc} \frac{\partial }{\partial x }& \frac{\partial }{\partial y }\\ A_y　B_z - A_z B_y & A_z　B_x - A_x B_z \end{array} \right| \\ &&\\ &=& {\bf e}_x \left( \frac{\partial (A_x B_y - A_y B_x)}{\partial y } - \frac{\partial (A_z B_x - A_x B_z)}{\partial z } \right) \\ && \quad + {\bf e}_y \left( \frac{\partial (A_y B_z - A_z B_y) }{\partial z } - \frac{\partial (A_x B_y - A_y B_x) }{\partial x } \right) \\ && \qquad + {\bf e}_z \left( \frac{\partial (A_z B_x - A_x B_z ) }{\partial x } - \frac{\partial (A_y B_z - A_z B_y) }{\partial y } \right) \\ &&\\ &=& \; {\bf e}_x \left( A_x \frac{\partial B_y}{\partial y } + \frac{\partial A_x}{\partial y } B_y - A_y \frac{\partial B_x}{\partial y } - \frac{\partial A_y}{\partial y } B_x \right) \\ && + {\bf e}_x \left( A_x \frac{\partial B_z}{\partial z } + \frac{\partial A_x}{\partial z } B_z - A_z \frac{\partial B_x}{\partial z } - \frac{\partial A_z}{\partial z } B_x \right) \\ && \quad + {\bf e}_y \left( A_y \frac{\partial B_z}{\partial z } + \frac{\partial A_y}{\partial z } B_z - A_z \frac{\partial B_y}{\partial z } - \frac{\partial A_z}{\partial z } B_y \right) \\ && \quad + {\bf e}_y \left( A_y \frac{\partial B_x}{\partial x } + \frac{\partial A_y}{\partial x } B_x - A_x \frac{\partial B_y}{\partial x } - \frac{\partial A_x}{\partial x } B_y \right) \\ && \qquad + {\bf e}_z \left( A_z \frac{\partial B_x}{\partial x } + \frac{\partial A_z}{\partial x } B_x - A_x \frac{\partial B_z}{\partial x } - \frac{\partial A_x}{\partial x } B_z \right) \\ && \qquad + {\bf e}_z \left( A_z \frac{\partial B_y}{\partial y } + \frac{\partial A_z}{\partial y } B_y - A_y \frac{\partial B_z}{\partial y } - \frac{\partial A_y}{\partial y } B_z \right) \\ &&\\ &=& \; {\bf e}_x \left( A_x \frac{\partial B_y}{\partial y } + A_x \frac{\partial B_z}{\partial z } - A_y \frac{\partial B_x}{\partial y } - A_z \frac{\partial B_x}{\partial z } \right) \\ && + {\bf e}_x \left( \frac{\partial A_x}{\partial y } B_y + \frac{\partial A_x}{\partial z } B_z - \frac{\partial A_y}{\partial y } B_x - \frac{\partial A_z}{\partial z } B_x \right) \\ && \quad + {\bf e}_y \left( A_y \frac{\partial B_x}{\partial x } + A_y \frac{\partial B_z}{\partial z } - A_x \frac{\partial B_y}{\partial x } - A_z \frac{\partial B_y}{\partial z } \right) \\ && \quad + {\bf e}_y \left( \frac{\partial A_y}{\partial x } B_x + \frac{\partial A_y}{\partial z } B_z - \frac{\partial A_x}{\partial x } B_y - \frac{\partial A_z}{\partial z } B_y \right) \\ && \qquad + {\bf e}_z \left( A_z \frac{\partial B_x}{\partial x } + A_z \frac{\partial B_y}{\partial y } - A_x \frac{\partial B_z}{\partial x } - A_y \frac{\partial B_z}{\partial y } \right) \\ && \qquad + {\bf e}_z \left( \frac{\partial A_z}{\partial x } B_x + \frac{\partial A_z}{\partial y } B_y - \frac{\partial A_x}{\partial x } B_z - \frac{\partial A_y}{\partial y } B_z \right) \\ &&\\ &=& \; {\bf e}_x \left( \underline{ A_x \frac{\partial B_x}{\partial x }} + A_x \frac{\partial B_y}{\partial y } + A_x \frac{\partial B_z}{\partial z } \underline{-A_x \frac{\partial B_x}{\partial x }} - A_y \frac{\partial B_x}{\partial y } - A_z \frac{\partial B_x}{\partial z } \right) \\ && + {\bf e}_x \left( \underline{ \frac{\partial A_x}{\partial x } B_x } + \frac{\partial A_x}{\partial y } B_y + \frac{\partial A_x}{\partial z } B_z \underline{-\frac{\partial A_x}{\partial x } B_x } - \frac{\partial A_y}{\partial y } B_x - \frac{\partial A_z}{\partial z } B_x \right) \\ && \quad + {\bf e}_y \left( A_y \frac{\partial B_x}{\partial x } \underline{ + A_y \frac{\partial B_y}{\partial y }} + A_y \frac{\partial B_z}{\partial z } - A_x \frac{\partial B_y}{\partial x } \underline{ - A_y \frac{\partial B_y}{\partial y }} - A_z \frac{\partial B_y}{\partial z } \right) \\ && \quad + {\bf e}_y \left( \frac{\partial A_y}{\partial x } B_x \underline{ + \frac{\partial A_y}{\partial y } B_y } + \frac{\partial A_y}{\partial z } B_z - \frac{\partial A_x}{\partial x } B_y \underline{ - \frac{\partial A_y}{\partial y } B_y } - \frac{\partial A_z}{\partial z } B_y \right) \\ && \qquad + {\bf e}_z \left( A_z \frac{\partial B_x}{\partial x } + A_z \frac{\partial B_y}{\partial y } \underline{ + A_z \frac{\partial B_z}{\partial z }} - A_x \frac{\partial B_z}{\partial x } - A_y \frac{\partial B_z}{\partial y } \underline{ - A_z \frac{\partial B_z}{\partial z }} \right) \\ && \qquad + {\bf e}_z \left( \frac{\partial A_z}{\partial x } B_x + \frac{\partial A_z}{\partial y } B_y \underline{ + \frac{\partial A_z}{\partial z } B_z } - \frac{\partial A_x}{\partial x } B_z - \frac{\partial A_y}{\partial y } B_z \underline{ - \frac{\partial A_z}{\partial z } B_z } \right) \\ &&\\ &=& {\bf e}_x \Big( A_x (\nabla \cdot {\bf B}) - ({\bf A} \cdot \nabla) B_x \Big) + {\bf e}_x \Big( ({\bf B} \cdot \nabla) A_x - B_x ( \nabla \cdot {\bf A}) \Big) \\ && \quad + {\bf e}_y \Big( A_y (\nabla \cdot {\bf B}) - ({\bf A} \cdot \nabla) B_y \Big) + {\bf e}_y \Big( ({\bf B} \cdot \nabla) A_y - B_y ( \nabla \cdot {\bf A}) \Big) \\ &&\qquad + {\bf e}_z \Big( A_z (\nabla \cdot {\bf B}) - ({\bf A} \cdot \nabla) B_z \Big) + {\bf e}_z \Big( ({\bf B} \cdot \nabla) A_z - B_x ( \nabla \cdot {\bf A}) \Big) \\ &&\\ &=& {\bf e}_x \Big( A_x (\nabla \cdot {\bf B}) - ({\bf A} \cdot \nabla) B_x + ({\bf B} \cdot \nabla) A_x - B_x ( \nabla \cdot {\bf A}) \Big) \\ && \quad + {\bf e}_y \Big( A_y (\nabla \cdot {\bf B}) - ({\bf A} \cdot \nabla) B_y + ({\bf B} \cdot \nabla) A_y - B_y ( \nabla \cdot {\bf A}) \Big) \\ &&\qquad + {\bf e}_z \Big( A_z (\nabla \cdot {\bf B}) - ({\bf A} \cdot \nabla) B_z + ({\bf B} \cdot \nabla) A_z - B_x ( \nabla \cdot {\bf A}) \Big) \\ &&\\ &=& {\bf A} (\nabla \cdot {\bf B}) - ({\bf A} \cdot \nabla) {\bf B} + ({\bf B} \cdot \nabla) {\bf A} - {\bf B} ( \nabla \cdot {\bf A}) \end{eqnarray*}

 

（９）

(9)   \begin{equation*} \nabla ({\bf A} \cdot {\bf B}) = ({\bf A} \cdot \nabla ){\bf B} + {\bf A} \times (\nabla \times {\bf B}) + ({\bf B} \cdot \nabla ){\bf A} + {\bf B} \times (\nabla \times {\bf A}) \end{equation*}

【 式（９）の証明を見る 】

式（９）の証明
式(9)左辺は，

    \begin{eqnarray*} && \nabla ({\bf A} \cdot {\bf B}) \\ &=& \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \Big( \left( A_x , A_y , A_z \right) \cdot \left( B_x , B_y , B_z \right) \Big) \\ &=& \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \left( A_x B_x + A_y B_y + A_z B_z \right) \\ &=& {\bf e}_x \frac{\partial }{\partial x } \left( A_x B_x + A_y B_y + A_z B_z \right) \\ && \quad +{\bf e}_y\frac{\partial }{\partial y } \left( A_x B_x + A_y B_y + A_z B_z \right) \\ && \qquad +{\bf e}_z\frac{\partial }{\partial z } \left( A_x B_x + A_y B_y + A_z B_z \right) \\ &=& {\bf e}_x \left( A_x \frac{\partial B_x}{\partial x } + \frac{\partial A_x }{\partial x } B_x + A_y \frac{\partial B_y}{\partial x } + \frac{\partial A_y }{\partial x } B_y + A_z \frac{\partial B_z}{\partial x } + \frac{\partial A_z }{\partial x } B_z \right) \\ && \quad + {\bf e}_y \left( A_x \frac{\partial B_x}{\partial y } + \frac{\partial A_x }{\partial y } B_x + A_y \frac{\partial B_y}{\partial y } + \frac{\partial A_y }{\partial y } B_y + A_z \frac{\partial B_z}{\partial y } + \frac{\partial A_z }{\partial y } B_z \right) \\ && \qquad + {\bf e}_z \left( A_x \frac{\partial B_x}{\partial z } + \frac{\partial A_x }{\partial z } B_x + A_y \frac{\partial B_y}{\partial z } + \frac{\partial A_y }{\partial z } B_y + A_z \frac{\partial B_z}{\partial z } + \frac{\partial A_z }{\partial z } B_z \right) \\ &=& {\bf e}_x \left( A_x \frac{\partial B_x}{\partial x } + A_y \frac{\partial B_y}{\partial x } + A_z \frac{\partial B_z}{\partial x } \right) \\ && \quad + {\bf e}_y \left( A_x \frac{\partial B_x}{\partial y } + A_y \frac{\partial B_y}{\partial y } + A_z \frac{\partial B_z}{\partial y } \right) \\ && \qquad + {\bf e}_z \left( A_x \frac{\partial B_x}{\partial z } + A_y \frac{\partial B_y}{\partial z }+ A_z \frac{\partial B_z}{\partial z } \right) \\ && \quad \qquad + {\bf e}_x \left( \frac{\partial A_x }{\partial x } B_x + \frac{\partial A_y }{\partial x } B_y + \frac{\partial A_z }{\partial x } B_z \right) \\ && \quad \qquad \quad + {\bf e}_y \left( \frac{\partial A_x }{\partial y } B_x + \frac{\partial A_y }{\partial y } B_y + \frac{\partial A_z }{\partial y } B_z \right) \\ && \quad \qquad \qquad + {\bf e}_z \left( \frac{\partial A_x }{\partial z } B_x + \frac{\partial A_y }{\partial z } B_y + \frac{\partial A_z }{\partial z } B_z \right) \\ \end{eqnarray*}

他方，式(9)右辺第1項は，

    \begin{eqnarray*} && ({\bf A} \cdot \nabla ){\bf B} \\ &=& \left( \left( A_x , A_y , A_z \right) \cdot \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \right) \left( B_x , B_y , B_z \right) \\ &=& \left( A_x \frac{\partial}{\partial x } + A_y \frac{\partial}{\partial y } + A_z \frac{\partial}{\partial z } \right) \left( B_x , B_y , B_z \right) \\ &=& {\bf e}_x \left( A_x \frac{\partial B_x}{\partial x } + A_y \frac{\partial B_x}{\partial y } + A_z \frac{\partial B_x}{\partial z } \right) \\ && \quad + {\bf e}_y \left( A_x \frac{\partial B_y}{\partial x } + A_y \frac{\partial B_y}{\partial y } + A_z \frac{\partial B_y}{\partial z } \right) \\ && \qquad + {\bf e}_z \left( A_x \frac{\partial B_z}{\partial x } + A_y \frac{\partial B_z}{\partial y }+ A_z \frac{\partial B_z}{\partial z } \right) \\ \end{eqnarray*}

式(9)右辺第2項は，

    \begin{eqnarray*} && {\bf A} \times (\nabla \times {\bf B}) \\ &=& \left( A_x , A_y , A_z \right) \times \left( \left( \frac{\partial}{\partial x },\frac{\partial}{\partial y },\frac{\partial}{\partial z } \right) \times \left( B_x , B_y , B_z \right)\right)\\ &=& \left( A_x , A_y , A_z \right) \times \left(　\frac{\partial B_z}{\partial y } - \frac{\partial B_y}{\partial z }, \; \frac{\partial B_x}{\partial z } - \frac{\partial B_z}{\partial x } \; \frac{\partial B_y}{\partial x } - \frac{\partial B_x}{\partial y } \right) \\ &=& {\bf e}_x \left[ A_y \left( \frac{\partial B_y}{\partial x } - \frac{\partial B_x}{\partial y } \right) - A_z \left( \frac{\partial B_x}{\partial z } - \frac{\partial B_z}{\partial x } \right) \right] \\ && \quad + {\bf e}_y \left[ A_z \left( \frac{\partial B_z}{\partial y } - \frac{\partial B_y}{\partial z } \right) - A_x \left( \frac{\partial B_y}{\partial x } - \frac{\partial B_x}{\partial y } \right) \right] \\ && \qquad + {\bf e}_z \left[ A_x \left( \frac{\partial B_x}{\partial z } - \frac{\partial B_z}{\partial x } \right) - A_y \left( \frac{\partial B_z}{\partial y } - \frac{\partial B_y}{\partial z } \right) \right] \\ &=& {\bf e}_x \left[ A_y \frac{\partial B_y}{\partial x } - A_y \frac{\partial B_x}{\partial y } - A_z \frac{\partial B_x}{\partial z } + A_z \frac{\partial B_z}{\partial x } \right] \\ && \quad + {\bf e}_y \left[ A_z \frac{\partial B_z}{\partial y } - A_z \frac{\partial B_y}{\partial z } - A_x \frac{\partial B_y}{\partial x } + A_x \frac{\partial B_x}{\partial y } \right] \\ && \qquad + {\bf e}_z \left[ A_x \frac{\partial B_x}{\partial z } - A_x \frac{\partial B_z}{\partial x } - A_y \frac{\partial B_z}{\partial y } + A_y \frac{\partial B_y}{\partial z } \right] \\ &=& {\bf e}_x \left[ A_y \frac{\partial B_y}{\partial x } + A_z \frac{\partial B_z}{\partial x } - A_y \frac{\partial B_x}{\partial y } - A_z \frac{\partial B_x}{\partial z } \right] \\ && \quad + {\bf e}_y \left[ A_x \frac{\partial B_x}{\partial y } + A_z \frac{\partial B_z}{\partial y } - A_x \frac{\partial B_y}{\partial x } - A_z \frac{\partial B_y}{\partial z } \right] \\ && \qquad + {\bf e}_z \left[ A_x \frac{\partial B_x}{\partial z } + A_y \frac{\partial B_y}{\partial z } - A_x \frac{\partial B_z}{\partial x } - A_y \frac{\partial B_z}{\partial y } \right] \\ \end{eqnarray*}

よって，式(9)右辺第1項＋右辺第2項は，

    \begin{eqnarray*} && ({\bf A} \cdot \nabla ){\bf B} + {\bf A} \times (\nabla \times {\bf B}) \\ &=& {\bf e}_x \left( A_x \frac{\partial B_x}{\partial x } \underline{ + A_y \frac{\partial B_x}{\partial y } + A_z \frac{\partial B_x}{\partial z }} \right) \\ && \quad + {\bf e}_y \left( \underline{ A_x \frac{\partial B_y}{\partial x } } + A_y \frac{\partial B_y}{\partial y } + \underline{ A_z \frac{\partial B_y}{\partial z } } \right) \\ && \qquad + {\bf e}_z \left( \underline{ A_x \frac{\partial B_z}{\partial x } + A_y \frac{\partial B_z}{\partial y } } + A_z \frac{\partial B_z}{\partial z } \right) \\ && + {\bf e}_x \left[ A_y \frac{\partial B_y}{\partial x } + A_z \frac{\partial B_z}{\partial x } \underline{- A_y \frac{\partial B_x}{\partial y } - A_z \frac{\partial B_x}{\partial z }} \right] \\ && \qquad + {\bf e}_y \left[ A_x \frac{\partial B_x}{\partial y } + A_z \frac{\partial B_z}{\partial y } \underline{- A_x \frac{\partial B_y}{\partial x } - A_z \frac{\partial B_y}{\partial z } } \right] \\ && \quad \qquad + {\bf e}_z \left[ A_x \frac{\partial B_x}{\partial z } + A_y \frac{\partial B_y}{\partial z } \underline{ - A_x \frac{\partial B_z}{\partial x } - A_y \frac{\partial B_z}{\partial y }} \right] \\ &=& {\bf e}_x \left( A_x \frac{\partial B_x}{\partial x } + A_y \frac{\partial B_y}{\partial x } + A_z \frac{\partial B_z}{\partial x } \right) \\ && \quad + {\bf e}_y \left( A_x \frac{\partial B_x}{\partial y } + A_y \frac{\partial B_y}{\partial y } + A_z \frac{\partial B_z}{\partial y } \right) \\ && \qquad + {\bf e}_z \left( A_x \frac{\partial B_x}{\partial z } + A_y \frac{\partial B_y}{\partial z }+ A_z \frac{\partial B_z}{\partial z } \right) \\ \end{eqnarray*}

となる．

同様にして，式(9)右辺第3項＋右辺第4項は，

    \begin{eqnarray*} && ({\bf B} \cdot \nabla ){\bf A} + {\bf B} \times (\nabla \times {\bf A}) \\ &=& {\bf e}_x \left( \frac{\partial A_x }{\partial x } B_x + \frac{\partial A_y }{\partial x } B_y + \frac{\partial A_z }{\partial x } B_z \right) \\ && \quad + {\bf e}_y \left( \frac{\partial A_x }{\partial y } B_x + \frac{\partial A_y }{\partial y } B_y + \frac{\partial A_z }{\partial y } B_z \right) \\ && \qquad + {\bf e}_z \left( \frac{\partial A_x }{\partial z } B_x + \frac{\partial A_y }{\partial z } B_y + \frac{\partial A_z }{\partial z } B_z \right) \\ \end{eqnarray*}

となる．

以上より，式(9)が成り立つことが示された．

 

３．次の公式が成り立つ．
（１０）

(10)   \begin{equation*} \nabla \times (\nabla \phi ) = {\rm rot} \; {\rm grad} \; \phi = 0 \end{equation*}

【 式（１０）の証明を見る 】

式（１０）の証明

    \begin{eqnarray*} &&\nabla \times (\nabla \phi ) \\ &=& \left( \frac{\partial }{\partial x }, \frac{\partial }{\partial y }, \frac{\partial}{\partial z } \right) \times \left( \frac{\partial }{\partial x }, \frac{\partial }{\partial y }, \frac{\partial}{\partial z } \phi \right)\\ &=& \left( \frac{\partial }{\partial x }, \frac{\partial }{\partial y }, \frac{\partial}{\partial z } \right) \times \left( \frac{\partial \phi }{\partial x }, \frac{\partial \phi }{\partial y }, \frac{\partial \phi }{\partial z } \right)\\ &=& \left| \begin{array}{ccc} {\bf e}_x & {\bf e}_y & {\bf e}_z\\ \frac{\partial }{\partial x }& \frac{\partial }{\partial y }& \frac{\partial}{\partial z }\\ \frac{\partial \phi }{\partial x } & \frac{\partial \phi }{\partial y } & \frac{\partial \phi }{\partial z } \end{array} \right| \\ &=& {\bf e}_x \left| \begin{array}{cc} \frac{\partial }{\partial y }& \frac{\partial}{\partial z }\\ \frac{\partial \phi }{\partial y } & \frac{\partial \phi }{\partial z } \end{array} \right| - {\bf e}_y \left| \begin{array}{cc} \frac{\partial }{\partial x }& \frac{\partial}{\partial z }\\ \frac{\partial \phi }{\partial x } & \frac{\partial \phi }{\partial z } \end{array} \right| + {\bf e}_z \left| \begin{array}{cc} \frac{\partial }{\partial x }& \frac{\partial}{\partial y } \\ \frac{\partial \phi }{\partial x } & \frac{\partial \phi }{\partial y } \end{array} \right| \\ &=& {\bf e}_x \left( \frac{\partial^2 \phi}{ \partial z \partial z } - \frac{\partial^2 \phi}{ \partial y \partial z } \right) \\ && \quad - {\bf e}_y \left( \frac{\partial^2 \phi}{ \partial z \partial x } - \frac{\partial^2 \phi}{ \partial x \partial z } \right) \\ && \qquad + {\bf e}_z \left( \frac{\partial^2 \phi}{ \partial y \partial x } - \frac{\partial^2 \phi}{ \partial x \partial y } \right) \\ &=& 0 \end{eqnarray*}

 

（１１）

(11)   \begin{equation*} \nabla \cdot (\nabla \times {\bf A} ) = {\rm div} \; {\rm rot} \; {\bf A} = 0 \end{equation*}

【 式（１１）の証明を見る 】

式（１１）の証明

    \begin{eqnarray*} &&\nabla \cdot ( \nabla \times {\bf A} ) \\ &=& \nabla \cdot \left( \left( \frac{\partial }{\partial x }, \frac{\partial }{\partial y }, \frac{\partial}{\partial z } \right) \times \left( A_x , A_y , A_z \right) \right)\\ &=& \nabla \cdot \left| \begin{array}{ccc} {\bf e}_x & {\bf e}_y & {\bf e}_z\\ \frac{\partial }{\partial x }& \frac{\partial }{\partial y }& \frac{\partial}{\partial z }\\ A_x & A_y & A_z \end{array} \right| \\ &=& \nabla \cdot \left( {\bf e}_x \left| \begin{array}{cc} \frac{\partial }{\partial y }& \frac{\partial}{\partial z }\\ A_y & A_z \end{array} \right| - {\bf e}_y \left| \begin{array}{cc} \frac{\partial }{\partial x }& \frac{\partial}{\partial z }\\ A_x & A_z \end{array} \right| + {\bf e}_z \left| \begin{array}{cc} \frac{\partial }{\partial x }& \frac{\partial}{\partial y } \\ A_x & A_y \end{array} \right| \right)\\ &=& \left( \frac{\partial }{\partial x }, \frac{\partial }{\partial y }, \frac{\partial}{\partial z } \right) \cdot \left( \frac{\partial A_z}{\partial y } - \frac{\partial A_y }{\partial z }, \frac{\partial A_x }{\partial z } - \frac{\partial A_z}{\partial x }, \frac{\partial A_y}{\partial x } - \frac{\partial A_x}{\partial y } \right) \\ &=& \frac{\partial }{\partial x } \left( \frac{\partial A_z}{\partial y } - \frac{\partial A_y }{\partial z } \right)+ \frac{\partial }{\partial y } \left( \frac{\partial A_x }{\partial z } - \frac{\partial A_z}{\partial x } \right)+\frac{\partial }{\partial z } \left( \frac{\partial A_y }{\partial x } - \frac{\partial A_x }{\partial y } \right) \\ &=& \left( \frac{\partial^2 A_z}{\partial x \partial y } - \frac{\partial^2 A_y }{\partial x \partial z } \right) + \left( \frac{\partial^2 A_x }{\partial y \partial z } - \frac{\partial^2 A_z}{ \partial y \partial x } \right) + \left( \frac{\partial^2 A_y }{\partial z \partial x } - \frac{\partial^2 A_x }{\partial z \partial y } \right) \\ &=& \frac{\partial^2 A_z}{\partial x \partial y } - \frac{\partial^2 A_z}{ \partial y \partial x } + \frac{\partial^2 A_y }{\partial z \partial x } - \frac{\partial^2 A_y }{\partial x \partial z } + \frac{\partial^2 A_x }{\partial y \partial z } - \frac{\partial^2 A_x }{\partial z \partial y } \\ &=& 0 \end{eqnarray*}

 

（１２）

(12)   \begin{equation*} \nabla \times (\nabla \times {\bf A}) = \nabla (\nabla \cdot {\bf A}) - \nabla^2 {\bf A} \end{equation*}

【 式（１２）の証明を見る 】

式（１２）の証明

    \begin{eqnarray*} && \nabla \times (\nabla \times {\bf A}) \\ &=& \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \times \left[ \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \times \left( A_x , A_y , A_z \right) \right] \\ &=& \left( \frac{\partial }{\partial x }, \frac{\partial }{\partial y }, \frac{\partial}{\partial z } \right) \times \left( \frac{\partial A_z}{\partial y } - \frac{\partial A_y }{\partial z }, \frac{\partial A_x }{\partial z } - \frac{\partial A_z}{\partial x }, \frac{\partial A_y}{\partial x } - \frac{\partial A_x}{\partial y } \right) \\ &=& \left| \begin{array}{ccc} {\bf e}_x & {\bf e}_y & {\bf e}_z\\ \frac{\partial }{\partial x }& \frac{\partial }{\partial y }& \frac{\partial}{\partial z }\\ \frac{\partial A_z}{\partial y } - \frac{\partial A_y }{\partial z }& \frac{\partial A_x }{\partial z } - \frac{\partial A_z}{\partial x }& \frac{\partial A_y}{\partial x } - \frac{\partial A_x}{\partial y } \end{array} \right| \\ &=& {\bf e}_x \left| \begin{array}{cc} \frac{\partial }{\partial y }& \frac{\partial}{\partial z }\\ \frac{\partial A_x }{\partial z } - \frac{\partial A_z}{\partial x }& \frac{\partial A_y}{\partial x } - \frac{\partial A_x}{\partial y } \end{array} \right| \\ && \quad - {\bf e}_y \left| \begin{array}{cc} \frac{\partial }{\partial x }& \frac{\partial}{\partial z }\\ \frac{\partial A_z}{\partial y } - \frac{\partial A_y }{\partial z }& \frac{\partial A_y}{\partial x } - \frac{\partial A_x}{\partial y } \end{array} \right| \\ && \qquad + {\bf e}_z \left| \begin{array}{cc} \frac{\partial }{\partial x }& \frac{\partial}{\partial y } \\ \frac{\partial A_z}{\partial y } - \frac{\partial A_y }{\partial z }& \frac{\partial A_x }{\partial z } - \frac{\partial A_z}{\partial x } \end{array} \right|\\ &=& {\bf e}_x \left(\frac{\partial^2 A_y}{\partial y \partial x } + \frac{\partial^2 A_z}{\partial z \partial x } - \frac{\partial^2 A_x}{\partial y^2 } - \frac{\partial^2 A_x }{\partial z^2 } \right)\\ && \quad + {\bf e}_y \left( \frac{\partial^2 A_z}{\partial z \partial y } + \frac{\partial^2 A_x}{\partial x \partial y } - \frac{\partial^2 A_y}{\partial x^2 } - \frac{\partial^2 A_y }{\partial z^2 } \right) \\ && \qquad + {\bf e}_z \left ( \frac{\partial^2 A_y }{\partial y \partial z } + \frac{\partial^2 A_x }{\partial x \partial z } - \frac{\partial^2 A_z}{\partial x^2 } - \frac{\partial^2 A_z}{\partial y^2 } \right)\\ &=& {\bf e}_x \left(\frac{\partial^2 A_y}{\partial y \partial x } + \frac{\partial^2 A_z}{\partial z \partial x } - \frac{\partial^2 A_x}{\partial y^2 } - \frac{\partial^2 A_x }{\partial z^2 } \underline{+ \frac{\partial^2 A_x}{\partial x^2 } - \frac{\partial^2 A_x}{\partial x^2 } } \right)\\ && \quad + {\bf e}_y \left( \frac{\partial^2 A_z}{\partial z \partial y } + \frac{\partial^2 A_x}{\partial x \partial y } - \frac{\partial^2 A_y}{\partial x^2 } - \frac{\partial^2 A_y }{\partial z^2 } \underline{ + \frac{\partial^2 A_y}{\partial y^2 } - \frac{\partial^2 A_y }{\partial y^2 } }\right)\\ && \qquad + {\bf e}_z \left ( \frac{\partial^2 A_y }{\partial y \partial z } + \frac{\partial^2 A_x }{\partial x \partial z } - \frac{\partial^2 A_z}{\partial x^2 } - \frac{\partial^2 A_z}{\partial y^2 } \underline{ + \frac{\partial^2 A_z}{\partial z^2 } - \frac{\partial^2 A_z}{\partial z^2 } } \right)\\ &=& {\bf e}_x \left(\frac{\partial^2 A_x}{\partial x^2 } + \frac{\partial^2 A_y}{\partial x \partial y } + \frac{\partial^2 A_z}{\partial x \partial z } \right)\\ && -{\bf e}_x \left(\frac{\partial^2 A_x}{\partial x^2 }+ \frac{\partial^2 A_x}{\partial y^2 } + \frac{\partial^2 A_x }{\partial z^2 } \right)\\ && \quad + {\bf e}_y \left( \frac{\partial^2 A_x}{\partial y \partial x } + \frac{\partial^2 A_y}{\partial y^2 } + \frac{\partial^2 A_z}{\partial y \partial z } \right)\\ && \quad - {\bf e}_y \left( \frac{\partial^2 A_y}{\partial x^2 } + \frac{\partial^2 A_y}{\partial y^2 } + \frac{\partial^2 A_y }{\partial z^2 } \right)\\ && \qquad + {\bf e}_z \left ( \frac{\partial^2 A_x }{\partial z \partial x } + \frac{\partial^2 A_y }{\partial z \partial y } + \frac{\partial^2 A_z}{\partial z^2 } \right)\\ && \qquad - {\bf e}_z \left (\frac{\partial^2 A_z}{\partial x^2 } + \frac{\partial^2 A_z}{\partial y^2 } + \frac{\partial^2 A_z}{\partial z^2 } } \right)\\ &=& {\bf e}_x \frac{\partial }{\partial x} \left(\frac{\partial A_x}{\partial x } + \frac{\partial A_y}{ \partial y } + \frac{\partial A_z}{ \partial z } \right)\\ && -{\bf e}_x \left(\frac{\partial^2 }{\partial x^2 }+ \frac{\partial^2 }{\partial y^2 } + \frac{\partial^2 }{\partial z^2 } \right)A_x\\ && \quad + {\bf e}_y \frac{\partial }{\partial y} \left( \frac{\partial A_x}{\partial x } + \frac{\partial A_y}{\partial y } + \frac{\partial A_z}{ \partial z } \right)\\ && \quad - {\bf e}_y \left( \frac{\partial^2 }{\partial x^2 } + \frac{\partial^2 }{\partial y^2 } + \frac{\partial^2 }{\partial z^2 } \right) A_y \\ && \qquad + {\bf e}_z \frac{\partial }{\partial z} \left( \frac{\partial A_x }{ \partial x } + \frac{\partial A_y }{ \partial y } + \frac{\partial A_z}{\partial z } \right)\\ && \qquad - {\bf e}_z \left(\frac{\partial^2 }{\partial x^2 } + \frac{\partial^2 }{\partial y^2 } + \frac{\partial^2 }{\partial z^2 } } \right)A_z\\ &=& {\bf e}_x \frac{\partial }{\partial x} \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \cdot \left( A_x , A_y , A_z \right)\\ && \quad + {\bf e}_y \frac{\partial }{\partial y} \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \cdot \left( A_x , A_y , A_z \right)\\ && \qquad + {\bf e}_z \frac{\partial }{\partial z} \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \cdot \left( A_x , A_y , A_z \right)\\ && -{\bf e}_x \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \cdot \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) A_x\\ && \quad - {\bf e}_y \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \cdot \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) A_y \\ && \qquad - {\bf e}_z \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \cdot \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right)A_z\\ &=& {\bf e}_x \frac{\partial }{\partial x} \left( \nabla \cdot {\bf A} \right) + {\bf e}_y \frac{\partial }{\partial y} \left( \nabla \cdot {\bf A} \right) + {\bf e}_z \frac{\partial }{\partial z} \left( \nabla \cdot {\bf A} \right)\\ && \quad -{\bf e}_x \left( \nabla \cdot \nabla \right) A_x - {\bf e}_y \left( \nabla \cdot \nabla \right) A_y - {\bf e}_z \left( \nabla \cdot \nabla \right) A_z\\ &=& \left( \frac{\partial}{\partial x }, \frac{\partial}{\partial y }, \frac{\partial}{\partial z } \right) \left( \nabla \cdot {\bf A} \right) - \left( \nabla \cdot \nabla \right) \left( A_x , A_y , A_z \right) \\ &=& \nabla \left( \nabla \cdot {\bf A} \right) - \nabla^2 {\bf A} \end{eqnarray*}

 

ベクトル解析の表記法

スカラー関数

スカラー値をとる関数を，スカラー関数という（これは，後述するベクトル関数との対比による）．記号として x(t)，y(t)，z(t)，\phi(t)，\psi(t) などを用いる．計算の途中などで明らかな場合，関数の引数（t）を省略して，単に x，y，z，\phi，\psi，などと書くことがある．

スカラー関数の積は \phi\psi，\phi(t)\psi(t) などと書く．

単位ベクトル（基底ベクトル）

単位ベクトル（あるいは，座標系の基底を定める基底ベクトル）の表記として，

(13)   \begin{equation*} {\bf e}_x =(1,0,0),\;{\bf e}_y =(0,1,0),\;{\bf e}_z =(0,0,1) \end{equation*}

あるいは

(14)   \begin{equation*} {\bf i} =(1,0,0),\;{\bf j} =(0,1,0),\;{\bf k} =(0,0,1) \end{equation*}

などを用いる．これにより，3次元ベクトル {\bf r} = (x,y,z) は，

(15)   \begin{equation*} {\bf r} =x{\bf e}_x +y{\bf e}_y +z{\bf e}_z \end{equation*}

あるいは

(16)   \begin{equation*} {\bf r} = x{\bf i} + y{\bf j} +z{\bf k} \end{equation*}

などとなる．

また，添字(index) i=1,2,3 を用いることにより，単位ベクトル および ベクトル {\bf r}=(x_1,x_2,x_3) を，

(17)   \begin{equation*} {\bf e}_1 =(1,0,0),\;{\bf e}_2 =(0,1,0),\;{\bf e}_3 =(0,0,1), \end{equation*}

(18)   \begin{eqnarray*} {\bf r} &=&x_1{\bf e}_1 +x_2{\bf e}_2 +x_3{\bf e}_3 \\ &=& \sum_{i=1}^3 x_i{\bf e}_i \end{eqnarray*}

などと書くこともできる．

なお，基底ベクトル {\bf e}_1,\;{\bf e}_2,\;{\bf e}_3 について，

(19)   \begin{equation*} {\bf e}_i\cdot {\bf e}_j = \delta_{ij} = \left\{ \begin{array}{cl} 1&(i=j)\\ 0&(i\not=j)\\ \end{array} \right \end{equation*}

となるとき，これを正規直交基底という．ただし，\delta_{ij} はクロネッカーのデルタである．

ベクトル関数

ベクトル {\bf A}=(A_x,A_y,A_z)，{\bf B}=(B_x,B_y,B_z) などの各成分が，引数 t の関数であるとき

(20)   \begin{eqnarray*} {\bf A}(t)&=& \left( A_x(t),A_y(t),A_z(t) \right), \\ {\bf B}(t)&=& \left( B_x(t),B_y(t),B_z(t) \right) \end{eqnarray*}

などと書き，これらをベクトル関数という．関数の引数（t）を省略して，単に {\bf A}，{\bf B}，などと書く場合がある．

内積

ベクトル解析では，横ベクトル（列ベクトル）表記が多用され，転置ベクトル(transposed vector)の記号 \;^t{\bf A}，{\bf A}^{\rm T}，{\bf A}^{\top} などはあまり用いられない（これは主に，冪指数や各次元の添字と転置記号の並記による煩雑を避けるとともに，計算スペースの都合上，横ベクトルを用いるため）．

ベクトルおよびベクトル関数の内積は，\cdot を用いて，単に {\bf A}\cdot{\bf B} などと書き，

(21)   \begin{equation*} {\bf A}\cdot{\bf B} = A_xB_x + A_yB_y + A_zB_z \end{equation*}

である．

微分演算子あるいは微分作用素(differential operator)

ナブラとラプラシアン

\nabla は，ナブラ(nabla)と読み，次式で定義される微分演算子(differential operator) である：

(22)   \begin{eqnarray*} \nabla &:=&\left( \frac{\partial}{\partial x },\frac{\partial}{\partial y },\frac{\partial}{\partial z } \right)\\ &=& {\bf e}_x\frac{\partial}{\partial x }+{\bf e}_y\frac{\partial}{\partial y} +{\bf e}_z\frac{\partial}{\partial z } \end{eqnarray*}

また，ナブラの内積

(23)   \begin{equation*} \Delta := \nabla \cdot \nabla = \frac{\partial^2}{\partial x^2 }+\frac{\partial ^2}{\partial y^2 }+\frac{\partial^2}{\partial z^2 } \end{equation*}

は，ラプラス演算子 あるいは ラプラシアン(Laplacian) という．

※なお，operator の訳語として，数学分野では「作用素」が，物理学分野では「演算子」が用いられることが多い．

勾配(grad)

スカラー関数に対する勾配(gradienat) {\rm grad} は，次式によって定義される．

(24)   \begin{eqnarray*} {\rm grad}\; \phi &:=& \nabla \phi \\ &=&\left( \frac{\partial \phi}{\partial x },\frac{\partial \phi}{\partial y },\frac{\partial \phi}{\partial z } \right)\\ &=& {\bf e}_x\frac{\partial \phi}{\partial x }+{\bf e}_y\frac{\partial \phi}{\partial y} +{\bf e}_z\frac{\partial \phi}{\partial z } \end{eqnarray*}

発散(div)

ベクトル関数に対する発散(divergence) {\rm div} は，次式によって定義される．

(25)   \begin{eqnarray*} {\rm div} {\bf A} &:=& \nabla \cdot {\bf A} \\ &=& \frac{\partial A_x}{\partial x }+\frac{\partial A_y}{\partial y }+\frac{\partial A_z}{\partial z } \end{eqnarray*}

すなわち，発散は，ナブラとベクトル関数の内積として与えられる．

回転(curl, rot)

ベクトル関数に対する回転(curl) {\rm curl} あるいは (rotation) {\rm rot} は，次式によって定義される．

(26)   \begin{eqnarray*} {\rm curl} {\bf A} &=& {\rm rot} {\bf A}\\ &:=& \nabla \times {\bf A} \\ &=& \left| \begin{array}{ccc} {\bf e}_x & {\bf e}_y & {\bf e}_z\\ \frac{\partial}{\partial x } & \frac{\partial}{\partial y } & \frac{\partial}{\partial z }\\ A_x & A_y & A_z \end{array} \right| \\ &=& {\bf e}_x \left| \begin{array}{cc} \frac{\partial}{\partial y } & \frac{\partial}{\partial z }\\ A_y & A_z \end{array} \right| - {\bf e}_y \left| \begin{array}{cc} \frac{\partial}{\partial x } & \frac{\partial}{\partial z }\\ A_x & A_z \end{array} \right| + {\bf e}_z \left| \begin{array}{cc} \frac{\partial}{\partial x } & \frac{\partial}{\partial y } \\ A_x & A_y \end{array} \right| \\ &=& {\bf e}_x \left( \frac{\partial A_z}{\partial y }-\frac{\partial A_y}{\partial z }\right) + {\bf e}_y \left(\frac{\partial A_x}{\partial z }-\frac{\partial A_z}{\partial x }\right)+{\bf e}_z \left(\frac{\partial A_y}{\partial x }-\frac{\partial A_x}{\partial y } \right)\\ &=& \left( \frac{\partial A_z}{\partial y }-\frac{\partial A_y}{\partial z }, \; \frac{\partial A_x}{\partial z }-\frac{\partial A_z}{\partial x },\;\frac{\partial A_y}{\partial x }-\frac{\partial A_x}{\partial y } \right) \end{eqnarray*}

すなわち，回転は，ナブラとベクトル関数のクロス積として与えられる．

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