ベクトルのいろいろな性質

ベクトル演算の性質を示す．以下，$\bm{A}$, $\bm{B}$, $\bm{C}$ は任意のベクトル場， $f$, $\phi$ は任意のスカラー場である．

• $\bm{A}\times\bm{B}=-\bm{B}\times\bm{A}$
• $\bm{A}\cdot(\bm{B}\times\bm{C})=\bm{B}\cdot(\bm{C}\times\bm{A})=\bm{C}(\bm{A}\times\bm{B})$
• $\bm{A}\times(\bm{B}\times\bm{C})=(\bm{A}\cdot\bm{C})\bm{B}-(\bm{A}\cdot\bm{B})\bm{C}$
• $\bm{A}\times\bm{B} = \bm{B}(\bm{B}\cdot\bm{C}) - \bm{C}(\bm{A}\cdot\bm{B})$
• $(\bm{A}+\bm{B}) \times (\bm{A}-\bm{B}) = 2(\bm{B}\times\bm{A})$
• $\nabla\times(\nabla \phi)=0$
• $\nabla\cdot(f\bm{A})=\nabla f\cdot\bm{A}+f\nabla\cdot\bm{A}$
  【証明】

  $$\nabla\cdot(f\bm{A}) =\left(\frac{\partial}{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)\cdot(fA_x,fA_y,fA_z)$$

  $$=\left(\frac{\partial f}{\partial x}A_x+f\frac{\partial A_x}{\par... ... \left(\frac{\partial f}{\partial z}A_z+f\frac{\partial A_z}{\partial z}\right)$$

  $$=\left(\frac{\partial f}{\partial x}A_x+\frac{\partial f}{\partia... ...ial x}+f\frac{\partial A_y}{\partial y}+f\frac{\partial A_z}{\partial z}\right)$$

  $$=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y... ...ac{\partial }{\partial y},\frac{\partial }{\partial z}\right)\cdot(A_x,A_y,A_z)$$

  $$=\nabla f\cdot\bm{A}+f\nabla\cdot\bm{A}$$

  
  

• $\nabla\cdot(\nabla\times\bm{A})=0$
  【証明】

  $$\nabla\cdot(\nabla\times\bm{A}) = \nabla\cdot\left(\frac{\partial}{\partial y}A_z - \frac{\partia... ... x}A_z,  \frac{\partial}{\partial x}A_y - \frac{\partial}{\partial y}A_x\right)$$

  $$= \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}A_z... ... z}\left(\frac{\partial}{\partial x}A_y - \frac{\partial}{\partial y}A_x\right)$$

  $$= \frac{\partial^2 A_z}{\partial x\partial y} - \frac{\partial^2 ... ...tial^2 A_y}{\partial z\partial x} - \frac{\partial^2 A_x}{\partial z\partial y}$$

  $$= \left(\frac{\partial^2 A_x}{\partial y\partial z} - \frac{\part... ...A_z}{\partial x\partial y} - \frac{\partial^2 A_z}{\partial y\partial x}\right)$$

  $$= 0$$

  
  

• $\nabla\cdot(\nabla\phi)=\nabla^2\phi$
• $\nabla\times(\nabla\times\bm{A})=\nabla(\nabla\cdot\bm{A})-\nabla^2\bm{A}$
• $\nabla\times(f\bm{A})=\nabla f\times\bm{A}+f\nabla\times\bm{A}$
  【証明】

  $$\nabla\times(f\bm{A)} =\left(\frac{\partial}{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)\times(fA_x,fA_y,fA_z)$$

  $$=\left(\frac{\partial f}{\partial y}A_z-\frac{\partial f}{\partia... ... f\frac{\partial A_z}{\partial y}-f\frac{\partial A_y}{\partial z}\right)\bm{i}$$

  $$\quad +\left(\frac{\partial f}{\partial z}A_x-\frac{\partial f}{\... ... f\frac{\partial A_x}{\partial z}-f\frac{\partial A_z}{\partial x}\right)\bm{j}$$

  $$\quad +\left(\frac{\partial f}{\partial x}A_y-\frac{\partial f}{\... ... f\frac{\partial A_y}{\partial x}-f\frac{\partial A_x}{\partial y}\right)\bm{k}$$

  $$=\left(\frac{\partial f}{\partial y}A_z-\frac{\partial f}{\partia... ...(\frac{\partial f}{\partial x}A_y-\frac{\partial f}{\partial y}A_x\right)\bm{k}$$

  $$\quad +\left(f\frac{\partial A_z}{\partial y}-f\frac{\partial A_y... ...(f\frac{\partial A_y}{\partial x}-f\frac{\partial A_x}{\partial y}\right)\bm{k}$$

  $$=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y... ...ac{\partial}{\partial y},\frac{\partial}{\partial z}\right) \times(A_x,A_y,A_z)$$

  $$=\nabla f\times\bm{A}+f\nabla\times\bm{A}$$

  
  

• $\nabla\cdot(\bm{A}\times\bm{B})=\bm{B}\cdot(\nabla\times\bm{A})-\bm{A}\cdot(\nabla\times\bm{B})$