ベクトル解析公式の証明 – 証明篇

すでに, クロネッカーのデルタやレヴィ=チヴィタ記号について成り立つ公式などはベクトル解析公式の証明 – 準備篇などを理解しているとして議論を進める.

そのなかでも特に重要な公式だけをあらためてまとめておこう. \[\begin{aligned} \delta_{ij} \mathrel{\mathop:}= \left\{ \begin{array}{ll} 1 & ( i = j ) \\ 0 & ( i \not = j) \\ \end{array} \right. \end{aligned}\] \[\begin{aligned} \epsilon_{ijk} \mathrel{\mathop:}= \left\{ \begin{array}{ll} 1 & (\text{$i$, $j$, $k$の偶置換}) \\ -1 & (\text{$i$, $j$, $k$の奇置換}) \\ 0 & (\text{それ以外}) \\ \end{array} \right. \quad. \end{aligned}\] \[ \begin{align} \epsilon_{ijk} \epsilon_{iqr} & = \delta_{{j}{q}} \delta_{{k}{r}} – \delta_{{j}{r}} \delta_{{k}{q}} \label{Levi-Civitaの積1} \\ \epsilon_{ijk} \epsilon_{ijr} & = 2\delta_{kr} \label{Levi-Civitaの積2} \\ \epsilon_{ijk} \epsilon_{ijk} & = 6 \label{Levi-Civitaの積3} \end{align} \]

また, 単項の中に二回以上同じ添字があらわれた時, その添字については和をとることを約束する, Einsteinの総和規約 \[A_{i} B_{i} = \sum_{i}^{} A_{i} B_{i}\] も用いることにする.

主なベクトル解析の公式の導出

\[\begin{aligned} & \boldsymbol{A} \cdot \left( \boldsymbol{B} \times \boldsymbol{C} \right) \\ \quad &= \sum_{i} \delta_{ij} A_i \sum_{k,l} \epsilon_{jkl} B_{k} C_{l} \\ &= \sum_{i,k,l} \epsilon_{ikl} A_i B_{k} C_{l} \\ &= \sum_{k,l} \epsilon_{1kl} A_1 B_{k} C_{l} + \sum_{k,l} \epsilon_{2kl} A_2 B_{k} C_{l} + \sum_{k,l} \epsilon_{3kl} A_3 B_{k} C_{l} \notag \\ &= \epsilon_{123} A_{1} B_{2} C_{3} + \epsilon_{132} A_{1} B_{3} C_{2} + \epsilon_{231} A_{2} B_{3} C_{1} \\ &+ \epsilon_{213} A_{2} B_{1} C_{3} + \epsilon_{312} A_{3} B_{1} C_{2} + \epsilon_{321} A_{3} B_{2} C_{1} \\ &= A_{1} B_{2} C_{3} – A_{1} B_{3} C_{2} + A_{2} B_{3} C_{1} \\ &- A_{2} B_{1} C_{3} + A_{3} B_{1} C_{2} – A_{3} B_{2} C_{1} \\ & = \left| \begin{array}{@{\,}ccc@{\,}} A_{1} & A_{2} & A_{3} \\ B_{1} & B_{2} & B_{2} \\ C_{1} & C_{2} & C_{3} \end{array} \right| \\ &= \boldsymbol{B} \cdot \left( \boldsymbol{C} \times \boldsymbol{A} \right) \\ &= \boldsymbol{C} \cdot \left( \boldsymbol{A} \times \boldsymbol{B} \right) \end{aligned}\]

\[\begin{aligned} & \left[ \boldsymbol{A} \times \left( \boldsymbol{B} \times \boldsymbol{C} \right) \right]_{i} \\ \quad &= \epsilon_{ijk} A_{j} \epsilon_{kqr} B_{q} C_{r} \\ &= \epsilon_{kij} \epsilon_{kqr} A_{j} B_{q} C_{r} \\ &= \left( {\delta_{{i}{q}} \delta_{{j}{r}} – \delta_{{i}{r}} \delta_{{j}{q}} } \right) A_{j} B_{q} C_{r} \\ &= \delta_{iq} \delta_{jr} A_{j} B_{q} C_{r} – \delta_{ir} \delta_{jq} A_{j} B_{q} C_{r} \notag \\ &= A_{j} B_{i} C_{j} – A_{j} B_{j} C_{i} \\ &= \left( \boldsymbol{A} \cdot \boldsymbol{C} \right) \boldsymbol{B}_{i} -\left( \boldsymbol{A} \cdot \boldsymbol{B} \right) \boldsymbol{C}_{i} \end{aligned}\]

\[\begin{aligned} & \left[ \boldsymbol{A} \times \left( \boldsymbol{B} \times \boldsymbol{C} \right) \right]_{i} + \left[ \boldsymbol{B} \times \left( \boldsymbol{C} \times \boldsymbol{A} \right) \right]_{i} + \left[ \boldsymbol{C} \times \left( \boldsymbol{A} \times \boldsymbol{B} \right) \right]_{i} \notag \\ &= \left( \boldsymbol{A} \cdot \boldsymbol{C} \right) \boldsymbol{B}_{i} -\left( \boldsymbol{A} \cdot \boldsymbol{B} \right) \boldsymbol{C}_{i} + \left( \boldsymbol{B} \cdot \boldsymbol{A} \right) \boldsymbol{C}_{i} \\ &-\left( \boldsymbol{B} \cdot \boldsymbol{C} \right) \boldsymbol{A}_{i} + \left( \boldsymbol{C} \cdot \boldsymbol{B} \right) \boldsymbol{A}_{i} -\left( \boldsymbol{C} \cdot \boldsymbol{A} \right) \boldsymbol{B}_{i} \\ &= \boldsymbol{0}_{i} \end{aligned}\]

\[\begin{aligned} & \left( \boldsymbol{A} \times \boldsymbol{B} \right) \cdot \left( \boldsymbol{C} \times \boldsymbol{D} \right) \\ \quad &= \delta_{ip} \left( \epsilon_{ijk} A_{j}B_{k} \right) \left( \epsilon_{pqr} C_{q} D_{r} \right) \\ &= \delta_{ip} \epsilon_{ijk} \epsilon_{pqr} A_{j}B_{k} C_{q} D_{r} \\ &= \epsilon_{ijk} \epsilon_{iqr} A_{j}B_{k} C_{q} D_{r} \\ &= \left( {\delta_{{j}{q}} \delta_{{k}{r}} – \delta_{{j}{r}} \delta_{{k}{q}} } \right) A_{j}B_{k} C_{q} D_{r} \\ &= \delta_{jq}\delta_{kr} A_{j}B_{k} C_{q} D_{r} – \delta_{jr}\delta_{kq} A_{j}B_{k} C_{q} D_{r} \\ &= A_{j}B_{k} C_{j} D_{k} – A_{j}B_{k} C_{k} D_{j} \\ &= \left( \boldsymbol{A} \cdot \boldsymbol{C} \right) \left( \boldsymbol{B} \cdot \boldsymbol{D} \right) – \left( \boldsymbol{B} \cdot \boldsymbol{C} \right) \left( \boldsymbol{A} \cdot \boldsymbol{D} \right) \end{aligned}\]

\[\begin{aligned} \boldsymbol{\nabla} \cdot \boldsymbol{\nabla} f &= \delta_{ij} \nabla_{i} \left( \nabla_{j} f \right) \\ &= \nabla_{i} \nabla_{i} f \\ &=\mathrel{\mathop:} \nabla^2f \\ &=\sum_{i} \frac{\partial^2 f }{\partial x_{i}^2} \end{aligned}\]

\[\begin{aligned} \left[ \boldsymbol{\nabla} \times \boldsymbol{\nabla} f \right]_{i} &= \epsilon_{ijk} \nabla_{j} \left( \nabla_{k} f \right) \\ &\underbrace{ = }_{ j \iff k } \epsilon_{ikj} \nabla_{k} \nabla_{j} f \\ &\underbrace{ = }_{\epsilon_{ikj} =- \epsilon_{ijk}} – \epsilon_{ijk} \nabla_{k} \nabla_{j} f \\ &=- \epsilon_{ijk} \nabla_{j} \nabla_{k} f \\ \to & \ \left[ \boldsymbol{\nabla} \times \boldsymbol{\nabla} f \right]_{i} = 0 \end{aligned}\]

\[\begin{aligned} \boldsymbol{\nabla} \cdot \left( \boldsymbol{\nabla} \times \boldsymbol{A} \right) &= \delta_{ij} \nabla_{i} \epsilon_{jkl} \nabla_{k}A_{l} \\ &= \epsilon_{jkl} \nabla_{j} \nabla_{k}A_{l} \\ &= – \epsilon_{jkl} \nabla_{j} \nabla_{k}A_{l} \\ \to &\ \boldsymbol{\nabla} \cdot \left( \boldsymbol{\nabla} \times \boldsymbol{A} \right) = 0 \end{aligned}\]

\[\begin{aligned} \left[ \boldsymbol{\nabla} \times \left( \boldsymbol{\nabla} \times \boldsymbol{A} \right) \right]_{i} &= \epsilon_{ijk} \left( \nabla_{j} \right) \left( \epsilon_{kqr} \nabla_{q} A_r \right) \\ &= \epsilon_{ijk} \epsilon_{kqr} \nabla_{j} \nabla_{q} A_r \\ &= \epsilon_{kij} \epsilon_{kqr} \nabla_{j} \nabla_{q} A_r \\ &= \left( {\delta_{{i}{q}} \delta_{{j}{r}} – \delta_{{i}{r}} \delta_{{j}{q}} } \right) \nabla_{j} \nabla_{q} A_r \\ &= \delta_{iq} \delta_{jr} \nabla_{j} \nabla_{q} A_r – \delta_{ir} \delta_{jq} \nabla_{j} \nabla_{q} A_r \\ &= \nabla_{j} \nabla_{i} A_j – \nabla_{j} \nabla_{j} A_i \\ &= \nabla_{i} \left( \boldsymbol{\nabla} \cdot \boldsymbol{A} \right) – \nabla^2 A_{i} \end{aligned}\]

\[\begin{aligned} \boldsymbol{\nabla} \left( f \boldsymbol{A} \right) &= \delta_{ij} \nabla_{i} \left( f A_{j} \right) \\ &= \delta_{ij} \left( \nabla_{i} f \right) A_j + \delta_{ij} f \left( \nabla_{i} A_{j} \right) \\ &= f\boldsymbol{\nabla} \cdot \boldsymbol{A} + \boldsymbol{A} \cdot \boldsymbol{\nabla} f \end{aligned}\]

\[\begin{aligned} \left[ \boldsymbol{\nabla} \times \left( f \boldsymbol{A} \right) \right]_{i} &= \epsilon_{ijk} \nabla_{j} \left( f A_{k} \right) \\ &= \epsilon_{ijk} \left( \nabla_{j} f \right) A_{k} + \epsilon_{ijk} f \left( \nabla_{j} A_{k} \right) \\ &= \epsilon_{ikj} \left( \nabla_{k} f \right) A_{j} + \epsilon_{ijk} f \left( \nabla_{j} A_{k} \right) \\ &= – \epsilon_{ijk} A_{j} \left( \nabla_{k} f \right) + \epsilon_{ijk} f \left( \nabla_{j} A_{k} \right) \\ &= \epsilon_{ijk} \left[ f \left( \nabla_{j} A_{k} \right) – A_{j} \left( \nabla_{k} f \right) \right] \\ &= \left[ f\boldsymbol{\nabla} \times \boldsymbol{A} – \boldsymbol{A} \times \boldsymbol{\nabla} f \right]_{i} \end{aligned}\]

\[\begin{aligned} \boldsymbol{\nabla} \cdot \left( \boldsymbol{A} \times \boldsymbol{B} \right) &= \delta_{ij} \nabla_{i} \left( \epsilon_{jkl} A_{k} B_{l} \right) \\ &= \delta_{ij} \epsilon_{jkl} \left\{ \left( \nabla_{i} A_{k} \right) B_{l} + A_{k} \left( \nabla_{i} B_{l} \right) \right\} \\ &= \epsilon_{ikl} \left\{ \left( \nabla_{i} A_{k} \right) B_{l} + A_{k} \left( \nabla_{i} B_{l} \right) \right\} \\ &= \epsilon_{lik} B_{l} \left( \nabla_{i} A_{k} \right) – \epsilon_{kil} A_{k} \left( \nabla_{i} B_{l} \right) \\ &= \epsilon_{ikl} B_{i} \left( \nabla_{k} A_{l} \right) – \epsilon_{ikl} A_{i} \left( \nabla_{k} B_{l} \right) \\ &= B_{i} \left( \epsilon_{ikl} \nabla_{k} A_{l} \right) – A_{i} \left( \epsilon_{ikl} \nabla_{k} B_{l} \right) \\ &= \boldsymbol{B} \cdot \left( \boldsymbol{\nabla} \times \boldsymbol{A} \right) – \boldsymbol{A} \cdot \left( \boldsymbol{\nabla} \times \boldsymbol{B} \right) \end{aligned}\]

\[\begin{aligned} \left[ \boldsymbol{\nabla} \times \left( \boldsymbol{A} \times \boldsymbol{B} \right) \right]_{i} &= \epsilon_{ijk} \nabla_{j} \left( \epsilon_{kqr} A_{q} B_{r} \right) \\ &= \epsilon_{kij} \epsilon_{kqr} \nabla_{j} \left( A_{q} B_{r} \right) \\ &= \left( {\delta_{{i}{q}} \delta_{{j}{r}} – \delta_{{i}{r}} \delta_{{j}{q}} } \right) \nabla_{j} \left( A_{q} B_{r} \right) \\ &= \delta_{iq} \delta_{jr} \nabla_{j} \left( A_{q} B_{r} \right) – \delta_{ir} \delta_{jq} \nabla_{j} \left( A_{q} B_{r} \right) \\ &= \nabla_{j} \left( A_{i} B_{j} \right) – \nabla_{j} \left( A_{j} B_{i} \right) \\ &= \left( \nabla_{j} A_{i} \right) B_{j} + A_{i} \left( \nabla_{j} B_{j} \right) – \left( \nabla_{j} A_{j} \right) B_{i} – A_{j} \left( \nabla_{j} B_{i} \right) \\ &= \left( \boldsymbol{B} \cdot \boldsymbol{\nabla} \right) A_{i} + A_{i} \left( \boldsymbol{\nabla} \cdot \boldsymbol{B} \right) – \left( \boldsymbol{\nabla} \cdot \boldsymbol{A} \right) B_{i} – \left( \boldsymbol{A} \cdot \boldsymbol{\nabla} \right) B_{i} \\ &= \left[ \left(\boldsymbol{B} \cdot \boldsymbol{\nabla} \right) \boldsymbol{A} – \left( \boldsymbol{A} \cdot \boldsymbol{\nabla} \right) \boldsymbol{B} – \boldsymbol{B} \left( \boldsymbol{\nabla} \cdot \boldsymbol{A} \right) + \boldsymbol{A} \left( \boldsymbol{\nabla} \cdot \boldsymbol{B} \right) \right]_{i} \end{aligned}\]

\[\begin{aligned} \boldsymbol{\nabla} \cdot \left( \boldsymbol{\nabla} f \times \boldsymbol{\nabla} g \right) &= \delta_{ij} \nabla_{i} \epsilon_{jkl} \left\{ \left( \nabla_{k} f \right) \left( \nabla_{l}g \right) \right\} \\ &= \underbrace{\epsilon_{ikl} \left( \nabla_{i} \nabla_{k} f \right) }_{=0} \left( \nabla_{l}g \right) + \left( \nabla_{k} f \right) \underbrace{\epsilon_{ikl} \left( \nabla_{i} \nabla_{l}g \right)}_{=0} \\ &=0 \end{aligned}\]

ベクトル解析の公式一覧

\[\begin{aligned} \boldsymbol{A} \cdot \left( \boldsymbol{B} \times \boldsymbol{C} \right) &= \left( \boldsymbol{A} \times \boldsymbol{B} \right) \cdot \boldsymbol{C} = \boldsymbol{B} \cdot \left( \boldsymbol{C} \times \boldsymbol{A} \right) = \boldsymbol{C} \cdot \left( \boldsymbol{B} \times \boldsymbol{A} \right)\\ &= \left| \begin{array}{@{\,}ccc@{\,}} A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \\ C_1 & C_2 & C_3 \end{array} \right| \end{aligned} \] \[ \boldsymbol{A} \times \left( \boldsymbol{B} \times \boldsymbol{C} \right) = \left( \boldsymbol{A} \cdot \boldsymbol{C} \right)\boldsymbol{B} – \left( \boldsymbol{A} \cdot \boldsymbol{B} \right)\boldsymbol{C} \] \[ \boldsymbol{A} \times \left( \boldsymbol{B} \times \boldsymbol{C} \right) + \boldsymbol{B} \times \left( \boldsymbol{C} \times \boldsymbol{A} \right) + \boldsymbol{C} \times \left( \boldsymbol{A} \times \boldsymbol{B} \right) = 0 \] \[ \left( \boldsymbol{A} \times \boldsymbol{B} \right) \cdot \left( \boldsymbol{C} \times \boldsymbol{D} \right) = \left( \boldsymbol{A} \cdot \boldsymbol{C} \right) \left( \boldsymbol{B} \cdot \boldsymbol{D} \right) – \left( \boldsymbol{B} \cdot \boldsymbol{C} \right) \left( \boldsymbol{A} \cdot \boldsymbol{D} \right) \] \[ \boldsymbol{\nabla} f = \mathrm{grad} \ f = \left( \frac{\partial f}{\partial x_{1}} , \frac{\partial f}{\partial x_{2}} , \frac{\partial f}{\partial x_{3}} \right) \] \[ \boldsymbol{\nabla} \cdot \boldsymbol{A} = \mathrm{div} \ \boldsymbol{A} = \frac{\partial A_{1}}{\partial x_{1}} + \frac{\partial A_{2}}{\partial x_{2}} + \frac{\partial A_{3}}{\partial x_{3}} \] \[ \boldsymbol{\nabla} \times \boldsymbol{A} = \mathrm{rot} \ \boldsymbol{A} = \left| \begin{array}{@{\,}ccc@{\,}} \boldsymbol{e}_{1} & \boldsymbol{e}_{2} & \boldsymbol{e}_{3} \\ \frac{\partial}{\partial x_{1}} & \frac{\partial}{\partial x_{2}} & \frac{\partial}{\partial x_{3}} \\ A_{1} & A_{2} & A_{3} \\ \end{array} \right| \] \[ \boldsymbol{\nabla} \cdot \boldsymbol{\nabla} f = \mathrm{div} \ \left( \mathrm{grad} \ f \right) = \nabla^2f = \frac{\partial^2 f }{\partial x_{1}^2} + \frac{\partial^2 f }{\partial x_{2}^2} + \frac{\partial^2 f }{\partial x_{3}^2} \] \[ \boldsymbol{\nabla} \times \boldsymbol{\nabla} f = \mathrm{rot} \left(\mathrm{grad} \ f \right) = \boldsymbol{0} \] \[ \boldsymbol{\nabla} \cdot \left( \boldsymbol{\nabla} \times \boldsymbol{A} \right) = \mathrm{div} \left( \mathrm{rot} \ \boldsymbol{A} \right) = 0 \] \[ \boldsymbol{\nabla} \times \left( \boldsymbol{\nabla} \times \boldsymbol{A} \right) = \mathrm{rot} \left(\mathrm{rot} \ \boldsymbol{A} \right) = \boldsymbol{\nabla} \left( \boldsymbol{\nabla} \cdot \boldsymbol{A} \right) – \nabla^2 \boldsymbol{A} \] \[ \boldsymbol{\nabla} \left( f \boldsymbol{A} \right) = f\boldsymbol{\nabla} \cdot \boldsymbol{A} + \boldsymbol{A} \cdot \boldsymbol{\nabla} f \] \[ \boldsymbol{\nabla} \times \left( f \boldsymbol{A} \right) = f\boldsymbol{\nabla} \times \boldsymbol{A} – \boldsymbol{A} \times \boldsymbol{\nabla} f \] \[ \boldsymbol{\nabla} \cdot \left( \boldsymbol{A} \times \boldsymbol{B} \right) = \boldsymbol{B} \cdot \left( \boldsymbol{\nabla} \times \boldsymbol{A} \right) – \boldsymbol{A} \cdot \left( \boldsymbol{\nabla} \times \boldsymbol{B} \right) \] \[ \boldsymbol{\nabla} \times \left( \boldsymbol{A} \times \boldsymbol{B} \right) = \left(\boldsymbol{B} \cdot \boldsymbol{\nabla} \right) \boldsymbol{A} – \left( \boldsymbol{A} \cdot \boldsymbol{\nabla} \right) \boldsymbol{B} – \boldsymbol{B} \left( \boldsymbol{\nabla} \cdot \boldsymbol{A} \right) + \boldsymbol{A} \left( \boldsymbol{\nabla} \cdot \boldsymbol{B} \right) \] \[ \boldsymbol{\nabla} \left( \boldsymbol{A} \cdot \boldsymbol{B} \right) = \boldsymbol{A} \times \left( \boldsymbol{\nabla} \times \boldsymbol{B} \right) + \left( \boldsymbol{A} \cdot \boldsymbol{\nabla} \right) \boldsymbol{B} + \boldsymbol{B} \times \left( \boldsymbol{\nabla} \times \boldsymbol{A} \right) + \left( \boldsymbol{B} \cdot \boldsymbol{\nabla} \right) \boldsymbol{A} \] \[ \boldsymbol{\nabla} \cdot \left( \boldsymbol{\nabla} f \times \boldsymbol{\nabla} g \right) = 0 \]

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