Benutzer:Zivilverteidigung/Spielwiese

$\mathrm {grad} \varphi (x_{1},\ldots ,x_{n})=\nabla \varphi (x_{1},\ldots ,x_{n})={\begin{pmatrix}{\frac {\partial \varphi }{\partial x_{1}}}\\\vdots \\{\frac {\partial \varphi }{\partial x_{n}}}\end{pmatrix}}$

$\operatorname {grad} \,c={\vec {0}}$
$\operatorname {grad} \,(c\cdot u)=c\cdot \operatorname {grad} \,u$
$\operatorname {grad} \,(u+v)=\operatorname {grad} \,u+\operatorname {grad} \,v$
$\operatorname {grad} \,(u\cdot v)=u\cdot \operatorname {grad} \,v+v\cdot \operatorname {grad} \,u$

$\operatorname {div} {\vec {F}}={\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}=\nabla \cdot {\vec {F}}$

$\operatorname {div} \,{\vec {c}}=0$
$\operatorname {div} \,c\cdot {\vec {F}}=c\cdot \operatorname {div} \,{\vec {F}}$
$\operatorname {div} \,({\vec {F}}+{\vec {G}})=\operatorname {div} \,{\vec {F}}+\operatorname {div} \,{\vec {G}}$
$\operatorname {div} \,(u\cdot {\vec {F}})=\operatorname {grad} \,u\cdot {\vec {F}}+u\cdot \operatorname {div} \,{\vec {F}}$
$\operatorname {div} ({\vec {F}}\times {\vec {G}})={\vec {G}}\,\operatorname {rot} \,{\vec {F}}-{\vec {F}}\,\operatorname {rot} \,{\vec {G}}$

$\operatorname {rot} {\vec {F}}=\nabla \times {\vec {F}}={\begin{pmatrix}{\frac {\partial }{\partial x}}\\{\frac {\partial }{\partial y}}\\{\frac {\partial }{\partial z}}\end{pmatrix}}\times {\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\end{pmatrix}}={\begin{pmatrix}{\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\\{\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\\{\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\end{pmatrix}}$

$\operatorname {rot} \,{\vec {c}}\cdot {\vec {F}}={\vec {c}}\cdot \operatorname {rot} \,{\vec {F}}$
$\operatorname {rot} \,({\vec {F}}+{\vec {G}})=\operatorname {rot} \,{\vec {F}}+\operatorname {rot} \,{\vec {G}}$
$\operatorname {rot} (u{\vec {F}})=u\,\operatorname {rot} \,{\vec {F}}+\operatorname {grad} \,u\times {\vec {F}}$
$\operatorname {rot} ({\vec {F}}\times {\vec {G}})=\left({\vec {G}}\cdot \operatorname {grad} \right){\vec {F}}-\left({\vec {F}}\cdot \operatorname {grad} \right){\vec {G}}+{\vec {F}}\,\operatorname {div} \,{\vec {G}}-{\vec {G}}\,\operatorname {div} \,{\vec {F}}$
$\operatorname {rot} \;\operatorname {grad} \,u={\vec {0}}$
$\operatorname {rot} \;(\operatorname {rot} \,u)=\operatorname {grad} (\operatorname {div} \,u)-\Delta u$