Question

A student asked me to prove\[\nabla\times\left(\vec F\times\vec G\right)=(\vec G\cdot\nabla)\vec F-(\vec F\cdot\nabla)\vec G+(\nabla\cdot\vec G)\vec F-(\nabla\cdot\vec F)\vec G,\]and I ended up having to stop midway through because of how long and tedious the process is. Would someone mind helping me find a proof of it? It's a well-known vector operator identity.

Answer 1

try khan academy under linear algebra section. Maybe it there.

Answer 2

Use \[\left[ \vec A\times \left[ \vec B\times \vec C \right] \right]=\vec B \cdot \left( \vec A \cdot \vec C \right) - \vec C \cdot \left( \vec A \cdot \vec B \right)\]

Answer 3

somebody seems to have made sense of this, though I can't complete this proof I don't think:
http://www.physicsforums.com/showthread.php?t=459660

Answer 4

Try use BAC-CAB and identity: \[\left[ \vec A \times\left[ \vec B \times \vec C \right]\right] +\left[ \vec C \times\left[ \vec A \times \vec B \right]\right]+\left[ \vec B \times\left[ \vec C \times \vec A \right]\right] =0\]

Answer 5

\[\vec \nabla \times (\vec F \times \vec G)= -\vec G \times (\vec \nabla \times \vec F) -\vec F \times (\vec G \times \vec \nabla)\]

Answer 6

|dw:1343807669966:dw|


\[ \hat i \left( {\partial \over \partial y} (F_xG_y - F_yG_x) - {\partial \over \partial z} (F_zG_x - F_xG_z) \right) + \\
\hat j \left( {\partial \over \partial z} (F_yG_z - F_zG_y) - {\partial \over \partial x} (F_xG_y - F_yG_x) \right) + \\
\hat k \left( {\partial \over \partial x} (F_zG_x- F_xG_z) - {\partial \over \partial y} (F_yG_z - F_zG_y) \right) \]

\[ =
\hat i F_x \left ( {\partial G_y \over \partial y} + {\partial G_z \over \partial z} \right ) + \hat j F_y \left ( {\partial G_z \over \partial z} + {\partial G_x \over \partial x} \right ) + \hat k F_z \left ( {\partial G_x \over \partial x} + {\partial G_y \over \partial y} \right ) \\
- \left (\hat i G_x \left ( {\partial F_y \over \partial y} + {\partial F_z \over \partial z} \right ) + \hat j G_y \left ( {\partial F_z \over \partial z} + {\partial F_x \over \partial x} \right ) + \hat k G_z \left ( {\partial F_x \over \partial x} + {\partial F_y \over \partial y} \right )\right ) \]

\[ =
\hat i F_x \left ( {\partial G_x \over \partial x} +{\partial G_y \over \partial y} + {\partial G_z \over \partial z} \right ) + \hat j F_y \left ( {\partial G_y \over \partial y}+ {\partial G_z \over \partial z} + {\partial G_x \over \partial x} \right )\\
+ \hat k F_z \left ( {\partial G_x \over \partial x} + {\partial G_y \over \partial y}+{\partial G_z \over \partial z} \right ) - \left( \hat i F_x \left( \partial G_x \over \partial x \right ) +\hat j F_y \left( \partial G_y \over \partial y \right ) + \hat k F_x \left( \partial G_z \over \partial z \right )\right )
- \\ \left (\hat i G_x \left ( {\partial F_y \over \partial y} + {\partial F_z \over \partial z} + {\partial F_x \over \partial x} \right ) + \hat j G_y \left ( {\partial F_z \over \partial z} + {\partial F_x \over \partial x} +{\partial F_y \over \partial y} \right ) + \hat k G_z \left ( {\partial F_x \over \partial x} + {\partial F_y \over \partial y} +{\partial F_z \over \partial z}\right ) \right ) \\
+ \left( \hat i G_x \left ( \partial F_x \over \partial x \right ) + \hat j G_y \left ( \partial F_y \over \partial y \right ) + \hat k G_z \left ( \partial F_z \over \partial z \right )\right ) \]
\[ \left ( \hat i F_x + \hat j F_y + \hat k F_z \right ) \left ( \nabla \cdot \vec G \right ) - \left ( \vec F \cdot \nabla \right )\vec G \\ -\left( \hat i G_x + \hat j G_y + \hat k F_z \right ) \left ( \nabla \cdot \vec F \right ) + \left( \vec G \cdot \nabla \right ) \vec F\]
hence the required result follows