2.2 Definition von grad, div, rot und $\Delta$ in verschiedenen Koordinatensystemen

Definition der Vektoroperatoren und deren Darstellung in den gebräuchlichsten Koordinatensystemen ^2.3

• Definition: Divergenz div

  $$\vec{A}\, \vec{r}\, \lim_{\Delta V \rightarrow 0}^{} {1 \over {\vert\Delta V\vert}} \oint_{\Delta F}^{} \vec{F}\, \vec{A}\, \oint_{\Delta F}^{} \vec{F}\, \vec{A}\,$$ (2.10)

• Definition: Rotation rot

  $${{\Delta F} \over {\vert\Delta F\vert}} \vec{A}\, \vec{r}\, \lim_{\Delta F \rightarrow 0}^{} \oint_{\Delta C}^{} \vec{r}\, \vec{A}\, \oint_{\Delta C}^{} \vec{r}\, \vec{A}\,$$ (2.11)

$$\Phi \left\{\vphantom{ \begin{array}{ll} \displaystyle {{\partial \Phi... ... \theta}} \vec{e_{\theta}} & \mbox{(Kugelkoordinaten)} \\ \end{array} }\right. \begin{array}{ll} \displaystyle {{\partial \Phi}\over {\partial x... ...{\partial \theta}} \vec{e_{\theta}} & \mbox{(Kugelkoordinaten)} \\ \end{array} \left.\vphantom{ \begin{array}{ll} \displaystyle {{\partial \Phi}... ... \theta}} \vec{e_{\theta}} & \mbox{(Kugelkoordinaten)} \\ \end{array} }\right.$$ (2.12)

$$\vec{A}\, \left\{\vphantom{ \begin{array}{l} \displaystyle {{\partial A_x}\... ...sin(\theta)}}{{\partial A_{\phi}}\over {\partial \phi}} \\ \end{array}}\right. \begin{array}{l} \displaystyle {{\partial A_x}\over {\partial x}}... ...ver {r \sin(\theta)}}{{\partial A_{\phi}}\over {\partial \phi}} \\ \end{array} \left.\vphantom{ \begin{array}{l} \displaystyle {{\partial A_x}\o... ...sin(\theta)}}{{\partial A_{\phi}}\over {\partial \phi}} \\ \end{array}}\right.$$ (2.13)

$$\vec{A}\, \left\{\vphantom{ \begin{array}{l} \displaystyle \left( {{\partia... ... A_r}\over {\partial \theta}} \right) \vec{e_{\phi}} \\ \\ \end{array}}\right. \begin{array}{l} \displaystyle \left( {{\partial A_z}\over {\part... ...\partial A_r}\over {\partial \theta}} \right) \vec{e_{\phi}} \\ \\ \end{array} \left.\vphantom{ \begin{array}{l} \displaystyle \left( {{\partial... ... A_r}\over {\partial \theta}} \right) \vec{e_{\phi}} \\ \\ \end{array}}\right.$$ (2.14)

$$\Delta \Phi \left\{\vphantom{ \begin{array}{l} \displaystyle {{\partial^2 \Ph... ...\theta)}} {{\partial^2 \Phi}\over {\partial \phi^2}} \\ \\ \end{array}}\right. \begin{array}{l} \displaystyle {{\partial^2 \Phi}\over {\partial ... ... \sin^2(\theta)}} {{\partial^2 \Phi}\over {\partial \phi^2}} \\ \\ \end{array} \left.\vphantom{ \begin{array}{l} \displaystyle {{\partial^2 \Phi... ...\theta)}} {{\partial^2 \Phi}\over {\partial \phi^2}} \\ \\ \end{array}}\right.$$ (2.15)