Vektoranalysis: Teil V

Anhang

In diesem Anhang stelle ich zunächst die Beweise der elementaren Rechengesetze für die Differentialoperatoren Gradient, Divergenz und Rotation zusammen. Dabei gehe ich auch die Anwendung des Differentialoperators Nabla ein. Danach beweise ich die Rechengesetze für die Kombinationen der Differentialoperatoren. Dabei setze ich die Rechengesetze der Analysis (skalerer Funktionen) als bekannt (und bewiesen) voraus.

Zur deutlichen und auffälligen Unterscheidung verwende ich dabei für skalare Ortsfunktionen die Buchstaben f = f(x, y, z), g = g(x, y, z) usw., für vektorielle Ortsfunktionen die Buchstaben v = v(x, y, z), w = w(x, y, z) usw.

Elementare Rechengesetze für die Differentialoperatoren

Gradient

${\begin{aligned}\mathrm {grad} (f+g)&\equiv {\vec {i}}{\frac {\partial (f+g)}{\partial x}}+{\vec {j}}{\frac {\partial (f+g)}{\partial y}}+{\vec {k}}{\frac {\partial (f+g)}{\partial z}}\\&={\vec {i}}{\frac {\partial f}{\partial x}}+{\vec {j}}{\frac {\partial f}{\partial y}}+{\vec {k}}{\frac {\partial f}{\partial z}}+{\vec {i}}{\frac {\partial g}{\partial x}}+{\vec {j}}{\frac {\partial g}{\partial y}}+{\vec {k}}{\frac {\partial g}{\partial z}}\\&=\mathrm {grad} (f)+\mathrm {grad(g)} \\\\\nabla (f+g)&=\nabla f+\nabla g\end{aligned}}$

${\begin{aligned}\mathrm {grad} (f\cdot g)&\equiv {\vec {i}}{\frac {\partial (f\cdot g)}{\partial x}}+{\vec {j}}{\frac {\partial (f\cdot g)}{\partial y}}+{\vec {k}}{\frac {\partial (f\cdot g)}{\partial z}}\\&={\vec {i}}{\frac {\partial f}{\partial x}}g+{\vec {i}}f{\frac {\partial g}{\partial x}}+{\vec {j}}{\frac {\partial f}{\partial y}}g+{\vec {j}}f{\frac {\partial g}{\partial y}}+{\vec {k}}{\frac {\partial f}{\partial z}}+{\vec {k}}f{\frac {\partial g}{\partial z}}\\&=\mathrm {grad} (f)g+\mathrm {f\,grad(g)} \\\\\nabla (f\cdot g)&=(\nabla f)g+f\nabla g\end{aligned}}$

${\begin{aligned}\mathrm {grad} (f^{n})&\equiv {\vec {i}}{\frac {\partial f^{n}}{\partial x}}+{\vec {j}}{\frac {\partial f}{\partial y}}^{n}+{\vec {k}}{\frac {\partial f^{n}}{\partial z}}\\&={\vec {i}}nf^{n-1}{\frac {\partial f}{\partial x}}+{\vec {j}}nf^{n-1}{\frac {\partial f}{\partial y}}+{\vec {k}}nf^{n-1}{\frac {\partial f}{\partial z}}\\&=nf^{n-1}\mathrm {grad} (f)\\&\\\nabla f^{n}&=nf^{n-1}\nabla f\end{aligned}}$

${\begin{aligned}\mathrm {grad} \,f(g)&\equiv {\vec {i}}{\frac {\partial f(g)}{\partial x}}+{\vec {j}}{\frac {\partial f(g)}{\partial y}}+{\vec {k}}{\frac {\partial f(g)}{\partial z}}\\&={\vec {i}}{\frac {\partial f(g)}{\partial g}}{\frac {\partial g}{\partial x}}+{\vec {j}}{\frac {\partial f(g)}{\partial g}}{\frac {\partial g}{\partial y}}+{\vec {k}}{\frac {\partial f(g)}{\partial g}}{\frac {\partial g}{\partial z}}\\&={\frac {\partial f(g)}{\partial g}}\mathrm {grad} (g)\\&\\\nabla f(g)&={\frac {\partial f(g)}{\partial g}}\nabla g\end{aligned}}$

Divergenz

${\begin{aligned}\mathrm {div} ({\vec {v}}+{\vec {w}})&\equiv {\frac {\partial ({\vec {v}}+{\vec {w}})_{x}}{\partial x}}+{\frac {\partial ({\vec {v}}+{\vec {w}})_{y}}{\partial y}}+{\frac {\partial ({\vec {v}}+{\vec {w}})_{z}}{\partial z}}\\&={\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}+{\frac {\partial w_{x}}{\partial x}}+{\frac {\partial w_{y}}{\partial y}}+{\frac {\partial w_{z}}{\partial z}}\\&=\mathrm {div} \,{\vec {v}}+\mathrm {div} \,{\vec {w}}\\&\\\nabla ({\vec {v}}+{\vec {w}})&=\nabla {\vec {v}}+\nabla {\vec {w}}\end{aligned}}$

${\begin{aligned}\mathrm {div} (f\cdot {\vec {v}})&\equiv {\frac {\partial (f\cdot {\vec {v}})_{x}}{\partial x}}+{\frac {\partial (f\cdot {\vec {v}})_{y}}{\partial y}}+{\frac {\partial (f\cdot {\vec {v}})_{z}}{\partial z}}\\&={\frac {\partial f}{\partial x}}{\vec {v}}_{x}+f{\frac {\partial v_{x}}{\partial x}}+{\frac {\partial f}{\partial y}}{\vec {v}}_{y}+f{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial f}{\partial z}}{\vec {v}}_{z}+f{\frac {\partial v_{z}}{\partial z}}\\&=(\mathrm {grad} \,f)\cdot {\vec {v}}+f\,\mathrm {div\,} {\vec {v}}\\&\\\nabla (f\cdot {\vec {v}})&=\nabla f\cdot {\vec {v}}+f\nabla {\vec {v}}\end{aligned}}$

${\begin{aligned}\mathrm {div} ({\vec {v}}\times {\vec {w}})&\equiv {\frac {\partial ({\vec {v}}\times {\vec {w}})_{x}}{\partial x}}+{\frac {\partial ({\vec {v}}\times {\vec {w}})_{y}}{\partial y}}+{\frac {\partial ({\vec {v}}\times {\vec {w}})_{z}}{\partial z}}\\&={\frac {\partial }{\partial x}}(v_{y}w_{z}-v_{z}w_{y})+{\frac {\partial }{\partial y}}(v_{z}w_{x}-v_{x}w_{z})+{\frac {\partial }{\partial z}}(v_{x}w_{y}-v_{y}w_{x})\\&={\frac {\partial v_{y}}{\partial x}}w_{z}+v_{y}{\frac {\partial w_{z}}{\partial x}}-{\frac {\partial v_{z}}{\partial x}}w_{y}-v_{z}{\frac {\partial w_{y}}{\partial x}}\\&+{\frac {\partial v_{z}}{\partial y}}w_{x}+v_{z}{\frac {\partial w_{x}}{\partial y}}-{\frac {\partial v_{x}}{\partial y}}w_{z}-v_{x}{\frac {\partial w_{z}}{\partial y}}\\&+{\frac {\partial v_{x}}{\partial z}}w_{y}+v_{x}{\frac {\partial w_{y}}{\partial z}}-{\frac {\partial v_{y}}{\partial z}}w_{x}-v_{y}{\frac {\partial w_{x}}{\partial z}}\\&=({\vec {\mathrm {rot} }}\,{\vec {v}}){\vec {w}}-{\vec {v}}\,{\vec {\mathrm {rot} }}\,{\vec {w}}\\&\\\nabla ({\vec {v}}\times {\vec {w}})&=(\nabla \times {\vec {v}}){\vec {w}}-{\vec {v}}(\nabla \times {\vec {w}})\end{aligned}}$

Rotation

${\begin{aligned}\mathrm {rot} ({\vec {v}}+{\vec {w}})&=\left|{\begin{array}{ccc}{\vec {i}}&{\vec {j}}&{\vec {k}}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\({\vec {v}}+{\vec {w}})_{x}&({\vec {v}}+{\vec {w}})_{y}&({\vec {v}}+{\vec {w}})_{z}\end{array}}\right|\\&=\left|{\begin{array}{ccc}{\vec {i}}&{\vec {j}}&{\vec {k}}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\v_{x}&v_{y}&v_{z}\end{array}}\right|+\left|{\begin{array}{ccc}{\vec {i}}&{\vec {j}}&{\vec {k}}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\w_{x}&w_{y}&w_{z}\end{array}}\right|\\&=\mathrm {rot} \,{\vec {v}}+\mathrm {rot} \,{\vec {w}}\\&\\\nabla \times ({\vec {v}}+{\vec {w}})&=\nabla \times {\vec {v}}+\nabla \times {\vec {w}}\end{aligned}}$

${\begin{aligned}\mathrm {rot} (f\cdot {\vec {v}})&=\left|{\begin{array}{ccc}{\vec {i}}&{\vec {j}}&{\vec {k}}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\f\cdot v_{x}&f\cdot v_{y}&f\cdot v_{z}\end{array}}\right|\\&={\vec {i}}\left({\frac {\partial }{\partial y}}f\cdot v_{z}-{\frac {\partial }{\partial z}}f\cdot v_{y}\right)+{\vec {j}}\left({\frac {\partial }{\partial z}}f\cdot v_{x}-{\frac {\partial }{\partial x}}f\cdot v_{z}\right)+{\vec {k}}\left({\frac {\partial }{\partial x}}f\cdot v_{y}-{\frac {\partial }{\partial y}}f\cdot v_{x}\right)\\&={\vec {i}}\left({\frac {\partial f}{\partial y}}v_{z}+f{\frac {\partial v_{z}}{\partial y}}-{\frac {\partial f}{\partial z}}v_{y}-f{\frac {\partial v_{y}}{\partial z}}\right)\\&+{\vec {j}}\left({\frac {\partial f}{\partial z}}v_{x}+f{\frac {\partial v_{x}}{\partial z}}-{\frac {\partial f}{\partial x}}v_{z}-f{\frac {\partial v_{z}}{\partial x}}\right)\\&+{\vec {k}}\left({\frac {\partial f}{\partial x}}v_{y}+f{\frac {\partial v_{y}}{\partial x}}-{\frac {\partial f}{\partial y}}v_{x}-f{\frac {\partial v_{x}}{\partial y}}\right)\\&=f\cdot \left[{\vec {i}}\left({\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}}\right)+{\vec {j}}\left({\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}}\right)+{\vec {k}}\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right)\right]\\&+{\vec {i}}\left({\frac {\partial f}{\partial y}}v_{z}-{\frac {\partial f}{\partial z}}v_{y}\right)+{\vec {j}}\left({\frac {\partial f}{\partial z}}v_{x}-{\frac {\partial f}{\partial x}}v_{z}\right)+{\vec {k}}\left({\frac {\partial f}{\partial x}}v_{y}-{\frac {\partial f}{\partial y}}v_{x}\right)\\&=f\,\mathrm {rot} \,{\vec {v}}+\left|{\begin{array}{ccc}{\vec {i}}&{\vec {j}}&{\vec {k}}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\v_{x}&v_{y}&v_{z}\end{array}}\right|\\&=f\,\mathrm {rot} \,{\vec {v}}+(\mathrm {grad} \,f)\times {\vec {v}}=(\mathrm {grad} \,f)\times {\vec {v}}+f\,\mathrm {rot} \,{\vec {v}}\\&\\\nabla \times (f\cdot {\vec {v}})&=\nabla f\times {\vec {v}}+f\cdot (\nabla \times {\vec {v}})\end{aligned}}$

Zweifache Differentialoperationen

Durch Kombination von zwei Differentialoperatoren (Gradient, Divergenz, Rotation) werden die zweiten partiellen Ableitungen der betreffenden Funktion(en) gebildet. Wegen der Natur (Skalar oder Vektor) und wegen der Anwendbarkeit (auf skalare Funktionen oder auf Vektorfunktionen) sind nur bestimmte Kombinationen möglich. Diese sind:

${\begin{aligned}\mathrm {grad} \,\mathrm {div} \,{\vec {v}}&=\nabla (\nabla {\vec {v}}),\\\mathrm {div} \,\mathrm {grad\,} f&=\nabla (\nabla f),\\\mathrm {div\,rot} \,{\vec {v}}&=\nabla (\nabla \times {\vec {v}}),\\\mathrm {rot\,grad\,} f&=\nabla \times (\mathrm {grad} \,f),\\\mathrm {rot\,rot\,} {\vec {v}}&=\nabla \times (\nabla \times {\vec {v}}).\end{aligned}}$

Wir untersuchen nun diese Kombinationen:

${\begin{aligned}\mathrm {grad\,div\,} {\vec {v}}&=\nabla (\nabla {\vec {v}})={\vec {i}}{\frac {\partial }{\partial x}}\mathrm {div} \,{\vec {v}}+{\vec {j}}{\frac {\partial }{\partial y}}\mathrm {div\,} {\vec {v}}+{\vec {k}}{\frac {\partial }{\partial z}}\mathrm {div} \,{\vec {v}}\\&={\vec {i}}\left({\frac {\partial ^{2}v_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{y}}{\partial y\partial x}}+{\frac {\partial ^{2}v_{z}}{\partial z\partial x}}\right)+{\vec {j}}\left({\frac {\partial ^{2}v_{x}}{\partial x\partial y}}+{\frac {\partial ^{2}v_{y}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{z}}{\partial z\partial y}}\right)\\&+{\vec {k}}\left({\frac {\partial ^{2}v_{x}}{\partial x\partial z}}+{\frac {\partial ^{2}v_{y}}{\partial y\partial z}}+{\frac {\partial ^{2}v_{z}}{\partial z^{2}}}\right)\end{aligned}}$

Weitere Vereinfachungen oder symbolische Abkürzungen sind nicht möglich.

$\mathrm {div\,grad\,} f=\nabla (\nabla f)=\mathrm {\mathrm {div} \left({\vec {i}}{\frac {\partial f}{\partial x}}+{\vec {j}}{\frac {\partial f}{\partial y}}+{\vec {k}}{\frac {\partial f}{\partial z}}\right)={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}} =\Delta f$

Wo $\Delta$ den LAPLACE-Operator bezeichnet.

Für die Anwendung auf eine skalare Ortsfunktion gilt also $\nabla \nabla =\Delta$ .

${\begin{aligned}\mathrm {div\,rot\,{\vec {v}}} &=\nabla (\nabla \times {\vec {v}})=\mathrm {div} \left[{\vec {i}}\left({\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}}\right)+{\vec {j}}\left({\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}}\right)+{\vec {k}}\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right)\right]\\&={\frac {\partial }{\partial x}}\left({\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}}\right)+{\frac {\partial }{\partial y}}\left({\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}}\right)+{\frac {\partial }{\partial z}}\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right)=0\end{aligned}}$

Beweis durch Ausrechnen und Beachtung des Satzes von SCHWARZ, der besagt, dass bei Stetigkeit der zweiten Ableitungen die Reihenfolge der Differentiationen beliebig ist. Man sieht, dass in diesem Fall der symbolische Vektor Nabla der Regel folgt, dass das Skalarprodukt zweier aufeinander senkrechter Vektoren null ist.

${\begin{aligned}\mathrm {rot\,grad\,} f&=\nabla \times (\nabla f)=\mathrm {rot} \left({\vec {i}}{\frac {\partial f}{\partial x}}+{\vec {j}}{\frac {\partial f}{\partial y}}+{\vec {k}}{\frac {\partial f}{\partial z}}\right)=\left|{\begin{array}{ccc}{\vec {i}}&{\vec {j}}&{\vec {k}}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\{\frac {\partial f}{\partial x}}&{\frac {\partial f}{\partial y}}&{\frac {\partial f}{\partial z}}\end{array}}\right|\\&={\vec {i}}\left({\frac {\partial ^{2}f}{\partial z\partial y}}-{\frac {\partial ^{2}f}{\partial y\partial z}}\right)+{\vec {j}}\left({\frac {\partial ^{2}f}{\partial x\partial z}}-{\frac {\partial ^{2}f}{\partial z\partial x}}\right)+{\vec {k}}\left({\frac {\partial ^{2}f}{\partial y\partial x}}-{\frac {\partial ^{2}f}{\partial x\partial y}}\right)=0\end{aligned}}$

Die Begründung ist dieselbe wie oben (Satz von SCHWARZ). Auch hier folgt der Nabla-Operator einem Rechengesetz der Vektoralgebra: Das Vektorprodukt zweier paralleler Vektoren ist null.

${\begin{aligned}\mathrm {rot\,rot\,} {\vec {v}}&=\nabla \times (\nabla \times {\vec {v}})=\mathrm {rot\left|{\begin{array}{ccc}{\vec {i}}&{\vec {j}}&{\vec {k}}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\v_{x}&v_{y}&v_{z}\end{array}}\right|} \\&=\mathrm {rot} \left[{\vec {i}}\left({\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}}\right)+{\vec {j}}\left({\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}}\right)+{\vec {k}}\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right)\right]\\&=\left|{\begin{array}{ccc}{\vec {i}}&{\vec {j}}&{\vec {k}}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\\left({\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}}\right)&\left({\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}}\right)&\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right)\end{array}}\right|\\&={\vec {i}}\left[{\frac {\partial }{\partial y}}\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right)-{\frac {\partial }{\partial z}}\left({\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}}\right)\right]\\&+{\vec {j}}\left[{\frac {\partial }{\partial z}}\left({\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}}\right)-{\frac {\partial }{\partial x}}\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right)\right]\\&+{\vec {k}}\left[{\frac {\partial }{\partial x}}\left({\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}}\right)-{\frac {\partial }{\partial y}}\left({\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}}\right)\right]\\&={\vec {i}}\left({\frac {\partial ^{2}v_{y}}{\partial x\partial y}}-{\frac {\partial ^{2}v_{x}}{\partial y^{2}}}-{\frac {\partial ^{2}v_{x}}{\partial z^{2}}}+{\frac {\partial ^{2}v_{z}}{\partial x\partial z}}\right)\\&+{\vec {j}}\left({\frac {\partial ^{2}v_{z}}{\partial y\partial z}}-{\frac {\partial ^{2}v_{y}}{\partial z^{2}}}-{\frac {\partial ^{2}v_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{x}}{\partial y\partial x}}\right)\\&+{\vec {k}}\left({\frac {\partial ^{2}v_{x}}{\partial z\partial x}}-{\frac {\partial ^{2}v_{z}}{\partial x^{2}}}-{\frac {\partial ^{2}v_{z}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{y}}{\partial z\partial y}}\right)\end{aligned}}$

Dieser Vektor kann so umgeformt werden, dass daraus ein Ausdruck wird, der sich später in der Elektrodynamik als sehr nützlich erweist: Zu seinen Komponenten werden jeweils drei Terme addiert, die am Ende wieder subtrahiert werden, sodass daraus die Differenz zweier Vektoren wird. Der erste davon ist der Vektor grad div v.

${\begin{aligned}\mathrm {rot\,rot\,} {\vec {v}}&={\vec {i}}\left({\frac {\partial ^{2}v_{y}}{\partial x\partial y}}-{\frac {\partial ^{2}v_{x}}{\partial y^{2}}}-{\frac {\partial ^{2}v_{x}}{\partial z^{2}}}+{\frac {\partial ^{2}v_{z}}{\partial x\partial z}}+{\underset {=A}{\underbrace {{\frac {\partial ^{2}v_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{x}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{x}}{\partial z^{2}}}} }}\right)\\&+{\vec {j}}\left({\frac {\partial ^{2}v_{z}}{\partial y\partial z}}-{\frac {\partial ^{2}v_{y}}{\partial z^{2}}}-{\frac {\partial ^{2}v_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{x}}{\partial y\partial x}}+{\underset {=B}{\underbrace {{\frac {\partial ^{2}v_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{y}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{y}}{\partial z^{2}}}} }}\right)\\&+{\vec {k}}\left({\frac {\partial ^{2}v_{x}}{\partial z\partial x}}-{\frac {\partial ^{2}v_{z}}{\partial x^{2}}}-{\frac {\partial ^{2}v_{z}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{y}}{\partial z\partial y}}+{\underset {=C}{\underbrace {{\frac {\partial ^{2}v_{z}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{z}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{z}}{\partial z^{2}}}} }}\right)\\&={\vec {i}}\left({\frac {\partial ^{2}v_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{y}}{\partial x\partial y}}+{\frac {\partial ^{2}v_{z}}{\partial x\partial z}}\right)+{\vec {j}}\left({\frac {\partial ^{2}v_{x}}{\partial y\partial x}}+{\frac {\partial ^{2}v_{y}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{z}}{\partial y\partial z}}\right)\\&+{\vec {k}}\left({\frac {\partial ^{2}v_{x}}{\partial z\partial x}}+{\frac {\partial ^{2}v_{y}}{\partial z\partial y}}+{\frac {\partial ^{2}v_{z}}{\partial z^{2}}}\right)\\&-{\vec {i}}\left({\underset {=A}{\underbrace {{\frac {\partial ^{2}v_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{x}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{x}}{\partial z^{2}}}} }}\right)-{\vec {j}}\left({\underset {=B}{\underbrace {{\frac {\partial ^{2}v_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{y}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{y}}{\partial z^{2}}}} }}\right)\\&-{\vec {k}}\left({\underset {=C}{\underbrace {{\frac {\partial ^{2}v_{z}}{\partial x^{2}}}+{\frac {\partial ^{2}v_{z}}{\partial y^{2}}}+{\frac {\partial ^{2}v_{z}}{\partial z^{2}}}} }}\right)\\&=\mathrm {grad\,div\,} {\vec {v}}-({\vec {i}}\Delta v_{x}+{\vec {j}}\Delta v_{y}+{\vec {k}}\Delta v_{z}).\end{aligned}}$

Der letzte Term ist der Vektor, der durch Anwendung des LAPLACE-Operators auf den Vektor v entsteht. Also ist

$\mathrm {rot\,rot\,} {\vec {v}}=\mathrm {grad\,div\,} {\vec {v}}-\Delta {\vec {v}}$

oder

$\nabla \times (\nabla \times {\vec {v}})=\nabla (\nabla \cdot {\vec {v}})-(\nabla \cdot \nabla ){\vec {v}}.$

Teil IV | Inhaltsverzeichnis