2.3 Gradient, Divergenz, Rotation, Nabla ()

grad ( $\displaystyle \phi$ $\displaystyle \psi$ ) $\displaystyle \equiv$ $\displaystyle \nabla$ ( $\displaystyle \phi$ $\displaystyle \psi$ ) = $\displaystyle \phi$ $\displaystyle \nabla$ $\displaystyle \psi$ + $\displaystyle \psi$ $\displaystyle \nabla$ $\displaystyle \phi$

(2.16)

div( $\displaystyle \phi$ $\displaystyle \vec{A}\,$ ) $\displaystyle \equiv$ $\displaystyle \nabla$ ^.( $\displaystyle \phi$ $\displaystyle \vec{A}\,$ ) = $\displaystyle \vec{A}\,$ ^. $\displaystyle \nabla$ $\displaystyle \phi$ + $\displaystyle \phi$ $\displaystyle \nabla$ ^. $\displaystyle \vec{A}\,$

(2.17)

rot( $\displaystyle \phi$ $\displaystyle \vec{A}\,$ ) $\displaystyle \equiv$ $\displaystyle \nabla$ x ( $\displaystyle \phi$ $\displaystyle \vec{A}\,$ ) = $\displaystyle \phi$ $\displaystyle \nabla$ x $\displaystyle \vec{A}\,$ - $\displaystyle \vec{A}\,$ x $\displaystyle \nabla$ $\displaystyle \phi$ (2.18)

div( $\displaystyle \vec{A}\,$ x $\displaystyle \vec{B}\,$ ) $\displaystyle \equiv$ $\displaystyle \nabla$ ^.( $\displaystyle \vec{A}\,$ x $\displaystyle \vec{B}\,$ ) = $\displaystyle \vec{B}\,$ ^.( $\displaystyle \nabla$ x $\displaystyle \vec{A}\,$ ) - $\displaystyle \vec{A}\,$ ^.( $\displaystyle \nabla$ x $\displaystyle \vec{B}\,$ ) (2.19)

rot( $\displaystyle \vec{A}\,$ x $\displaystyle \vec{B}\,$ ) $\displaystyle \equiv$ $\displaystyle \nabla$ x ( $\displaystyle \vec{A}\,$ x $\displaystyle \vec{B}\,$ ) = $\displaystyle \vec{A}\,$ ( $\displaystyle \nabla$ ^. $\displaystyle \vec{B}\,$ ) - $\displaystyle \vec{B}\,$ ( $\displaystyle \nabla$ ^. $\displaystyle \vec{A}\,$ ) + ( $\displaystyle \vec{B}\,$ ^. $\displaystyle \nabla$ ) $\displaystyle \vec{A}\,$ - ( $\displaystyle \vec{A}\,$ ^. $\displaystyle \nabla$ ) $\displaystyle \vec{B}\,$ (2.20)

grad ( $\displaystyle \vec{A}\,$ ^. $\displaystyle \vec{B}\,$ ) $\displaystyle \equiv$ $\displaystyle \nabla$ ( $\displaystyle \vec{A}\,$ ^. $\displaystyle \vec{B}\,$ ) = $\displaystyle \vec{A}\,$ x ( $\displaystyle \nabla$ x $\displaystyle \vec{B}\,$ ) + $\displaystyle \vec{B}\,$ x ( $\displaystyle \nabla$ x $\displaystyle \vec{A}\,$ ) + ( $\displaystyle \vec{B}\,$ ^. $\displaystyle \nabla$ ) $\displaystyle \vec{A}\,$ + ( $\displaystyle \vec{A}\,$ ^. $\displaystyle \nabla$ ) $\displaystyle \vec{B}\,$ (2.21)

$\displaystyle \nabla^{2}_{}$ $\displaystyle \phi$ = $\displaystyle \nabla$ ^. $\displaystyle \nabla$ $\displaystyle \phi$ = $\displaystyle \Delta$ $\displaystyle \phi$

(2.22)

$\displaystyle \nabla^{2}_{}$ $\displaystyle \vec{A}\,$ = $\displaystyle \nabla$ ( $\displaystyle \nabla$ ^. $\displaystyle \vec{A}\,$ ) - $\displaystyle \nabla$ x ( $\displaystyle \nabla$ x $\displaystyle \vec{A}\,$ ) (2.23)