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.
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}
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:
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{\displaystyle {\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}}}
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{\displaystyle {\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.
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{\displaystyle \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
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{\displaystyle \Delta }
Für die Anwendung auf eine skalare Ortsfunktion gilt also
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{\displaystyle \nabla \nabla =\Delta }
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{\displaystyle {\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.
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}
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{\displaystyle {\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}}}
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{\displaystyle \mathrm {rot\,rot\,} {\vec {v}}=\mathrm {grad\,div\,} {\vec {v}}-\Delta {\vec {v}}}
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{\displaystyle \nabla \times (\nabla \times {\vec {v}})=\nabla (\nabla \cdot {\vec {v}})-(\nabla \cdot \nabla ){\vec {v}}.}
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f711f2eab70f1e6fcf85ea717ea00a43ae729097)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e62e71136398f15917191b6759afe24f711bdb)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/703c0a9610f6dcdea60a3d4cce98430dc6b8db9a)
