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2.3 Gradient, Divergenz, Rotation, Nabla ($ \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)

$\displaystyle \nabla$ x $\displaystyle \nabla$$\displaystyle \phi$ = 0 (2.24)

$\displaystyle \nabla$ . ($\displaystyle \nabla$ x $\displaystyle \vec{A}\,$) = 0 (2.25)


next up previous contents
Next: 2.4 Verschiedene Beziehungen des Up: 2. Vektoranalysis, Feldtheorie Previous: 2.2 Definition von grad,   Inhalt
Alexander Wagner
2000-04-14