Ing Mathematik: Funktionen mehrerer Veränderlicher

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Zusammenfassung

Betrachtet man Funktionen, die von mehreren Ortskoordinaten abhängen, so kann man sie nach jeder dieser Ortskoordinaten ableiten und das ggfs. auch mehrfach. Einige Linearkombinationen solcher Ableitungen werden besonders häufig verwendet, z. B der Gradient, die Divergenz, der Laplace-Operator oder der Drehimpulsoperator. Zusammenfassend bezeichnet man diese als Differentialoperatoren. Man kann Ortskoordinaten in verschiedenen Koordinatensystemen angeben. Häufig verwendet werden kartesische Koordinaten, Zylinderkoordinaten und Kugelkoordinaten. Betrachtet man dieselbe Funktion dargestellt in unterschiedlichen Koordinatensystemen, so sieht die Funktionsgleichung meist sehr unterschiedlich aus, je nachdem, in welchem Koordinatensystem man sie darstellt. Genauso sehen die Differentialoperatoren in unterschiedlichen Koordinatensystem unterschiedlich aus. Im Folgenden geben wir an, wie einige Differentialoperatoren in verschiedenen Koordinatensystemen aussehen und anschließend rechnen wir die angegebenen Formeln nach.

Partielle Ableitung

Siehe Ing Mathematik: Differenziation im Rn und  Partielle Ableitung.

Übungen:

• Sei $f(x,y)=x^{2}+y^{3}$. Berechne ${\frac {\partial f}{\partial x}}$ und ${\frac {\partial f}{\partial y}}$
• Sei $f(x,y)=\cos(x^{3}y^{4})$. Berechne ${\frac {\partial f}{\partial x}}$, ${\frac {\partial f}{\partial y}}$ und ${\frac {\partial ^{2}f}{\partial x^{2}}}$

Satz von Schwarz:

Hermann Amandus Schwarz (1843-1921), deutscher Mathematiker

Der Satz von Schwarz besagt, dass bei mehrfach stetig differenzierbaren Funktionen mehrerer Variablen die Reihenfolge, in der die partiellen Differentiationen (Ableitungen) nach den einzelnen Variablen durchgeführt werden, nicht entscheidend für das Ergebnis ist (von  Satz von Schwarz zitiert). D.h. ${\frac {\partial ^{2}f}{\partial x\partial y}}={\frac {\partial ^{2}f}{\partial y\partial x}}$. Ist die Funktion nicht stetig, so muss das nicht gelten (siehe z.B.  Satz_von_Schwarz#Gegenbeispiel)!

Übungen:

• Sei $f(x,y)=x^{3}\sin y$. Berechne ${\frac {\partial ^{2}f}{\partial x\partial y}}$ und ${\frac {\partial ^{2}f}{\partial y\partial x}}$.
• Sei $f(x,y)={\frac {\log |x|}{y}}$. Berechne ${\frac {\partial ^{2}f}{\partial x\partial y}}$ und ${\frac {\partial ^{2}f}{\partial y\partial x}}$.

Gradient

In kartesischen Koordinaten ist der Gradient folgendermaßen definiert:

$\mathbf {\text{grad}} \ f=\left({\begin{matrix}{\frac {\partial f}{\partial x}}\\{\frac {\partial f}{\partial y}}\\{\frac {\partial f}{\partial z}}\end{matrix}}\right)$

Übung: Sei $f(x,y,z)=x^{2}y+\sin z$. Berechne $\mathbf {\text{grad}} \ f$.

Das totale Differenzial lautet:

$\mathrm {d} f={\frac {\partial f}{\partial x}}\mathrm {d} x+{\frac {\partial f}{\partial y}}\mathrm {d} y+{\frac {\partial f}{\partial z}}\mathrm {d} z=\mathbf {\text{grad}} \ f\;\mathrm {d} \mathbf {r} \quad$ mit $\quad \mathrm {d} \mathbf {r} =\left({\begin{matrix}\mathrm {d} x\\\mathrm {d} y\\\mathrm {d} z\end{matrix}}\right)$

Der Nabla-Operator sei:

$\nabla =\left({\begin{matrix}{\frac {\partial }{\partial x}}\\{\frac {\partial }{\partial y}}\\{\frac {\partial }{\partial z}}\end{matrix}}\right)$

Da der Nabla-Operator ein symbolischer Vektor ist, wird er manchmal auch fett oder mit einem Pfeil geschrieben: ${\boldsymbol {\nabla }},\;{\vec {\nabla }}$. Eingeführt wurde er von Hamilton.

Sir William Rowan Hamilton (1805 - 1865) irischer Mathematiker und Physiker

Gradient und Nabla-Operator hängen somit folgendermaßen zusammen

$\mathbf {\text{grad}} \ f=\nabla f$

Pierre-Simon Laplace (1749-1827) französischer Mathematiker und Physiker

Der Laplace-Operator sei

$\Delta =\nabla \nabla =\nabla ^{2}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}$

Divergenz

Wir betrachten eine Funktion $\mathbf {f} \colon \,\mathbb {R} ^{3}\to \mathbb {R} ^{3}$. In Komponentenschreibweise ist $\mathbf {f}$ gegeben durch:

$\mathbf {f} =\left({\begin{matrix}f_{\mathrm {x} }\\f_{\mathrm {y} }\\f_{\mathrm {z} }\end{matrix}}\right)$

Es ist zu beachten, dass wir $\mathbf {f}$ fett (bold) gesetzt haben, wobei wir $f_{\mathrm {x} }$ normal (regular) gesetzt haben. Hierdurch drückt man in der Regel aus, dass $\mathbf {f}$ in einen mehrdimensionalen Vektorraum abbildet, bzw, dass $\mathbf {f}$ mehr als eine Komponente hat oder, anders ausgedrückt, $\mathbf {f}$ eine vektorwertige Funktion ist. Hingegen wird $f_{\mathrm {x} }$ normal gesetzt, weil es nur genau eine Komponente hat. Man beachte auch, dass hier mit $f_{\mathrm {x} }$ nicht die partielle Ableitung gemeint ist. Diese wird auch oft als ${\frac {\partial f}{\mathrm {d} x}}=f_{x}$ geschrieben. Eine andere Möglichkeit, die Vektorwertigkeit einer Funktion auszudrücken, ist ein Pfeil über dem Funktionssymbol:

${\vec {f}}$

Eine weitere Möglichkeit der Komponentenschreibweise ist:

$\mathbf {f} =\left({\begin{matrix}f_{1}\\f_{2}\\f_{3}\end{matrix}}\right)$

Abkürzend hierfür schreibt man auch:

$f_{i}$

Hierbei haben wir alle drei Komponenten der Funktion $\mathbf {f}$ zusammenfassend durch das Symbol $f_{i}$ ausgedrückt. Hierbei fällt auf, dass der Index $i$ im Symbol $f_{i}$ kursiv gesetzt wurde. Hingegen wurde der Index $\mathrm {z}$ im Symbol $f_{\mathbf {z} }$ normal gesetzt. Auch dies ist eine Konvention. Die Vektorwertigkeit von $f_{i}$ wird durch die Kursivschrift des $i$ ausgedrückt.

Die Divergenz in kartesischen Koordinaten ist definiert durch:

$\mathrm {div} \mathbf {f} =\nabla \cdot \mathbf {f} ={\frac {\partial f_{\mathrm {x} }}{\partial \mathrm {x} }}+{\frac {\partial f_{\mathrm {y} }}{\partial \mathrm {y} }}+{\frac {\partial f_{\mathrm {z} }}{\partial \mathrm {z} }}$

Eine alternative Schreibweise ist:

$\sum _{i=1}^{3}{\frac {\partial f_{i}}{\partial \mathrm {x} _{i}}}$

Diese kann man noch weiter verkürzen zu:

$\partial _{i}f_{i}$

Übung: Sei $\mathbf {f} =\left({\begin{matrix}x\\y\\z\end{matrix}}\right)$. Berechne $\mathrm {div} \mathbf {f}$.

Zylinderkoordinaten

Umrechnung von Zylinderkoordinaten in kartesische Koordinaten

Die Zylinderkoordinaten werden durch folgende Gleichungen definiert:

${\begin{matrix}x&=&\rho \cos(\phi )\\y&=&\rho \sin(\phi )\\z&=&z\end{matrix}}$

Umrechnung von kartesischen Koordinaten in Zylinderkoordinaten

Aus den Definitionsgleichungen erhält man:

${\begin{matrix}\rho ={\sqrt {x^{2}+y^{2}}}&=&\left(x^{2}+y^{2}\right)^{\frac {1}{2}}\\\phi &=&\mathrm {arctan} ({\frac {y}{x}})\end{matrix}}$

Übung: Leiten Sie die Formeln für $\rho$ und $\phi$ her.

Ableitungen der Zylinderkoordinaten nach den kartesischen Koordinaten

Leitet man die obigen Gleichungen ab, so erhält man:

${\frac {\partial \rho }{\partial x}}={\frac {1}{2}}(x^{2}+y^{2})^{-{\frac {1}{2}}}\cdot 2x={\frac {1}{\sqrt {x^{2}+y^{2}}}}x={\frac {x}{\rho }}=\cos(\phi )$

${\frac {\partial \rho }{\partial y}}={\frac {1}{2}}(x^{2}+y^{2})^{-{\frac {1}{2}}}\cdot 2y={\frac {y}{\rho }}=\sin(\phi )$

${\frac {\partial \phi }{\partial x}}={\frac {1}{1+\left({\frac {y}{x}}\right)^{2}}}\cdot {\frac {-y}{x^{2}}}=-{\frac {y}{\rho ^{2}}}=-{\frac {\sin(\phi )}{\rho }}$

${\frac {\partial \phi }{\partial y}}={\frac {1}{1+\left({\frac {y}{x}}\right)^{2}}}\cdot {\frac {1}{x}}={\frac {x}{\rho ^{2}}}={\frac {\cos(\phi )}{\rho }}$

Ableitung einer Funktion in Zylinderkoordinaten nach kartesischen Koordinaten

Will man eine Funktion in Zylinderkoordinaten nach kartesischen Koordinaten ableiten, so muss man die (mehrdimensionale) Kettenregel berücksichtigen und erhält:

${\frac {\partial }{\partial x}}f(\rho ,\phi ,z)={\frac {\partial \rho }{\partial x}}{\frac {\partial }{\partial \rho }}f(\rho ,\phi ,z)+{\frac {\partial \phi }{\partial x}}{\frac {\partial }{\partial \phi }}f(\rho ,\phi ,z)=\left(\cos(\phi ){\frac {\partial }{\partial \rho }}-{\frac {\sin(\phi )}{\rho }}{\frac {\partial }{\partial \phi }}\right)f(\rho ,\phi ,z)$

${\frac {\partial }{\partial y}}f(\rho ,\phi ,z)={\frac {\partial \rho }{\partial y}}{\frac {\partial }{\partial \rho }}f(\rho ,\phi ,z)+{\frac {\partial \phi }{\partial y}}{\frac {\partial }{\partial \phi }}f(\rho ,\phi ,z)=\left(\sin(\phi ){\frac {\partial }{\partial \rho }}+{\frac {\cos(\phi )}{\rho }}{\frac {\partial }{\partial \phi }}\right)f(\rho ,\phi ,z)$

Ableitungen der kartesischen Koordinaten nach den Zylinderkoordinaten

${\frac {\partial \mathbf {x} }{\partial \rho }}={\begin{pmatrix}\cos(\phi )\\\sin(\phi )\\0\end{pmatrix}};\;{\frac {\partial \mathbf {x} }{\partial \phi }}=\rho {\begin{pmatrix}-\sin(\phi )\\\cos(\phi )\\0\end{pmatrix}};\;{\frac {\partial \mathbf {x} }{\partial z}}=\rho {\begin{pmatrix}0\\0\\1\end{pmatrix}}$

Normiert:

${\boldsymbol {\hat {\rho }}}={\frac {\frac {\partial \mathbf {x} }{\partial \rho }}{\left|{\frac {\partial \mathbf {x} }{\partial \rho }}\right|}}={\begin{pmatrix}\cos(\phi )\\\sin(\phi )\\0\end{pmatrix}}$

${\boldsymbol {\hat {\phi }}}={\frac {\frac {\partial \mathbf {x} }{\partial \phi }}{\left|{\frac {\partial \mathbf {x} }{\partial \phi }}\right|}}={\begin{pmatrix}-\sin(\phi )\\\cos(\phi )\\0\end{pmatrix}}$

${\boldsymbol {\hat {z}}}={\frac {\frac {\partial \mathbf {x} }{\partial z}}{\left|{\frac {\partial \mathbf {x} }{\partial z}}\right|}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}$

Gradient:

$\nabla f(r,\phi ,z)=\left({\boldsymbol {\hat {\rho }}}{\frac {\partial }{\partial \rho }}+{\boldsymbol {\hat {\phi }}}{\frac {1}{\rho }}{\frac {\partial }{\partial \phi }}+{\boldsymbol {\hat {z}}}{\frac {\partial }{\partial z}}\right)f(r,\phi ,z)$

Divergenz:

$\mathbf {A} :={\begin{pmatrix}A_{x}\\A_{y}\\A_{z}\end{pmatrix}}={\begin{pmatrix}A_{\rho }\cos(\phi )-A_{\phi }\sin(\phi )\\A_{\rho }\sin(\phi )+A_{\phi }\cos(\phi )\\A_{z}\end{pmatrix}}$

${\frac {\partial }{\partial x}}A_{x}=\left({\frac {x}{\rho }}{\frac {\partial }{\partial \rho }}-{\frac {y}{\rho ^{2}}}{\frac {\partial }{\partial \phi }}\right)\left(A_{\rho }\cos(\phi )-A_{\phi }\sin(\phi )\right)={\frac {x}{\rho }}{\frac {\partial }{\partial \rho }}A_{\rho }\cos(\phi )+{\frac {y}{\rho ^{2}}}{\frac {\partial }{\partial \phi }}A_{\phi }\sin(\phi )-{\frac {x}{\rho }}{\frac {\partial }{\partial \rho }}A_{\phi }\sin(\phi )-{\frac {y}{\rho ^{2}}}{\frac {\partial }{\partial \phi }}A_{\rho }\cos(\phi )$

${\frac {\partial }{\partial x}}A_{x}={\frac {x}{\rho }}\cos(\phi ){\frac {\partial A_{\rho }}{\partial \rho }}+{\frac {y}{\rho ^{2}}}\sin(\phi ){\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {y}{\rho ^{2}}}A_{\phi }\cos(\phi )-{\frac {x}{\rho }}\sin(\phi ){\frac {\partial A_{\phi }}{\partial \rho }}-{\frac {y}{\rho ^{2}}}\cos(\phi ){\frac {\partial A_{\rho }}{\partial \phi }}+{\frac {y}{\rho ^{2}}}A_{\rho }\sin(\phi )$

${\frac {\partial }{\partial x}}A_{x}=\cos(\phi )^{2}{\frac {\partial A_{\rho }}{\partial \rho }}+{\frac {1}{\rho }}\sin(\phi )^{2}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {1}{\rho }}A_{\phi }\cos(\phi )\sin(\phi )-\cos(\phi )\sin(\phi ){\frac {\partial A_{\phi }}{\partial \rho }}-{\frac {1}{\rho }}\sin(\phi )\cos(\phi ){\frac {\partial A_{\rho }}{\partial \phi }}+{\frac {1}{\rho }}A_{\rho }\sin(\phi )^{2}$

${\frac {\partial }{\partial y}}A_{y}=\left({\frac {y}{\rho }}{\frac {\partial }{\partial \rho }}+{\frac {x}{\rho ^{2}}}{\frac {\partial }{\partial \phi }}\right)\left(A_{\rho }\sin(\phi )+A_{\phi }\cos(\phi )\right)={\frac {y}{\rho }}{\frac {\partial }{\partial \rho }}A_{\rho }\sin(\phi )+{\frac {x}{\rho ^{2}}}{\frac {\partial }{\partial \phi }}A_{\phi }\cos(\phi )+{\frac {y}{\rho }}{\frac {\partial }{\partial \rho }}A_{\phi }\cos(\phi )+{\frac {x}{\rho ^{2}}}{\frac {\partial }{\partial \phi }}A_{\rho }\sin(\phi )$

${\frac {\partial }{\partial y}}A_{y}={\frac {y}{\rho }}\sin(\phi ){\frac {\partial A_{\rho }}{\partial \rho }}-{\frac {x}{\rho ^{2}}}A_{\phi }\sin(\phi )+{\frac {x}{\rho ^{2}}}\cos(\phi ){\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {y}{\rho }}\cos(\phi ){\frac {\partial A_{\phi }}{\partial \rho }}+{\frac {x}{\rho ^{2}}}\sin(\phi ){\frac {\partial A_{\rho }}{\partial \phi }}+{\frac {x}{\rho ^{2}}}A_{\rho }\cos(\phi )$

${\frac {\partial }{\partial y}}A_{y}=\sin(\phi )^{2}{\frac {\partial A_{\rho }}{\partial \rho }}-{\frac {1}{\rho }}A_{\phi }\cos(\phi )\sin(\phi )+{\frac {1}{\rho }}\cos(\phi )^{2}{\frac {\partial A_{\phi }}{\partial \phi }}+\cos(\phi )\sin(\phi ){\frac {\partial A_{\phi }}{\partial \rho }}+{\frac {1}{\rho }}\cos(\phi )\sin(\phi ){\frac {\partial A_{\rho }}{\partial \phi }}+{\frac {1}{\rho }}A_{\rho }\cos(\phi )^{2}$

${\frac {\partial }{\partial x}}A_{x}+{\frac {\partial }{\partial y}}A_{y}={\frac {\partial A_{\rho }}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {1}{\rho }}A_{\rho }={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\rho A_{\rho }+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}$

$\nabla \cdot \mathbf {A} ={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\rho A_{\rho }+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {\partial A_{z}}{\partial z}}$

Sphärische Koordinaten

${\begin{matrix}x&=&r\sin(\Theta )\cos(\phi )\\y&=&r\sin(\Theta )\sin(\phi )\\z&=&r\cos(\Theta )\end{matrix}}$

${\begin{matrix}r&=&{\sqrt {x^{2}+y^{2}+z^{2}}}=\left(x^{2}+y^{2}+z^{2}\right)^{\frac {1}{2}}\\\phi &=&\mathrm {arctan} ({\frac {y}{x}})\\\Theta &=&\mathrm {arctan} ({\frac {\sqrt {x^{2}+y^{2}}}{z}})\end{matrix}}$

${\frac {\partial r}{\partial x}}={\frac {1}{2}}(x^{2}+y^{2}+z^{2})^{-{\frac {1}{2}}}\cdot 2x={\frac {x}{r}}=\sin(\Theta )\cos(\phi )$

${\frac {\partial r}{\partial y}}={\frac {y}{r}}=\sin(\Theta )\sin(\phi )$

${\frac {\partial r}{\partial z}}={\frac {z}{r}}=\cos(\Theta )$

$\rho :={\sqrt {x^{2}+y^{2}}}=r\sin(\Theta )$

${\frac {\partial \phi }{\partial x}}={\frac {1}{1+\left({\frac {y}{x}}\right)^{2}}}\cdot -{\frac {y}{x^{2}}}=-{\frac {y}{\rho ^{2}}}=-{\frac {\sin(\phi )}{\rho }}=-{\frac {\sin(\phi )}{r\sin(\Theta )}}$

${\frac {\partial \phi }{\partial y}}={\frac {1}{1+\left({\frac {y}{x}}\right)^{2}}}\cdot {\frac {1}{x}}={\frac {x}{\rho ^{2}}}={\frac {\cos(\phi )}{\rho }}={\frac {\cos(\phi )}{r\sin(\Theta )}}$

${\frac {\partial \phi }{\partial z}}=0$

${\frac {\partial \Theta }{\partial x}}={\frac {1}{1+{\frac {x^{2}+y^{2}}{z^{2}}}}}\cdot {\frac {2x}{2z{\sqrt {x^{2}+y^{2}}}}}={\frac {z}{r^{2}}}\cdot \cos(\phi )={\frac {1}{r}}\cdot \cos(\Theta )\cos(\phi )$

${\frac {\partial \Theta }{\partial y}}={\frac {1}{1+{\frac {x^{2}+y^{2}}{z^{2}}}}}\cdot {\frac {2y}{2z{\sqrt {x^{2}+y^{2}}}}}={\frac {z}{r^{2}}}\cdot \sin(\phi )={\frac {1}{r}}\cdot \cos(\Theta )\sin(\phi )$

${\frac {\partial \Theta }{\partial z}}={\frac {1}{1+{\frac {x^{2}+y^{2}}{z^{2}}}}}\cdot {\frac {-{\sqrt {x^{2}+y^{2}}}}{z^{2}}}=-{\frac {\rho }{r^{2}}}=-{\frac {\sin(\Theta )}{r}}$

${\frac {\partial \mathbf {x} }{\partial r}}={\begin{pmatrix}\sin(\Theta )\cos(\phi )\\\sin(\Theta )\sin(\phi )\\\cos(\Theta )\end{pmatrix}}$

${\frac {\partial \mathbf {x} }{\partial \phi }}={\begin{pmatrix}-r\sin(\Theta )\sin(\phi )\\r\sin(\Theta )\cos(\phi )\\0\end{pmatrix}}$

${\frac {\partial \mathbf {x} }{\partial \Theta }}={\begin{pmatrix}r\cos(\Theta )\cos(\phi )\\r\cos(\Theta )\sin(\phi )\\-r\sin(\Theta )\end{pmatrix}}$

${\boldsymbol {\hat {r}}}={\frac {\frac {\partial \mathbf {x} }{\partial r}}{|{\frac {\partial \mathbf {x} }{\partial r}}|}}={\begin{pmatrix}\sin(\Theta )\cos(\phi )\\\sin(\Theta )\sin(\phi )\\\cos(\Theta )\end{pmatrix}}$

${\boldsymbol {\hat {\phi }}}={\frac {\frac {\partial \mathbf {x} }{\partial \phi }}{|{\frac {\partial \mathbf {x} }{\partial \phi }}|}}={\begin{pmatrix}-\sin(\phi )\\\cos(\phi )\\0\end{pmatrix}}$

${\boldsymbol {\hat {\Theta }}}={\frac {\frac {\partial \mathbf {x} }{\partial \Theta }}{|{\frac {\partial \mathbf {x} }{\partial \Theta }}|}}={\begin{pmatrix}\cos(\Theta )\cos(\phi )\\\cos(\Theta )\sin(\phi )\\-\sin(\Theta )\end{pmatrix}}$

${\frac {\partial }{\partial x}}f(r,\Theta ,\phi )=\left({\frac {\partial r}{\partial x}}{\frac {\partial }{\partial r}}+{\frac {\partial \phi }{\partial x}}{\frac {\partial }{\partial \phi }}+{\frac {\partial \Theta }{\partial x}}{\frac {\partial }{\partial \Theta }}\right)f(r,\Theta ,\phi )=\left(\sin(\Theta )\cos(\phi ){\frac {\partial }{\partial r}}-{\frac {\sin(\phi )}{r\sin(\Theta )}}{\frac {\partial }{\partial \phi }}+{\frac {1}{r}}\cdot \cos(\Theta )\cos(\phi ){\frac {\partial }{\partial \Theta }}\right)f(r,\Theta ,\phi )$

${\frac {\partial }{\partial y}}f(r,\Theta ,\phi )=\left({\frac {\partial r}{\partial y}}{\frac {\partial }{\partial r}}+{\frac {\partial \phi }{\partial y}}{\frac {\partial }{\partial \phi }}+{\frac {\partial \Theta }{\partial y}}{\frac {\partial }{\partial \Theta }}\right)f(r,\Theta ,\phi )=\left(\sin(\Theta )\sin(\phi ){\frac {\partial }{\partial r}}+{\frac {\cos(\phi )}{r\sin(\Theta )}}{\frac {\partial }{\partial \phi }}+{\frac {1}{r}}\cdot \cos(\Theta )\sin(\phi ){\frac {\partial }{\partial \Theta }}\right)f(r,\Theta ,\phi )$

${\frac {\partial }{\partial z}}f(r,\Theta ,\phi )=\left({\frac {\partial r}{\partial z}}{\frac {\partial }{\partial r}}+{\frac {\partial \phi }{\partial z}}{\frac {\partial }{\partial \phi }}+{\frac {\partial \Theta }{\partial z}}{\frac {\partial }{\partial \Theta }}\right)f(r,\Theta ,\phi )=\left(\cos(\Theta ){\frac {\partial }{\partial r}}-{\frac {1}{r}}\cdot \sin(\Theta ){\frac {\partial }{\partial \Theta }}\right)f(r,\Theta ,\phi )$

$\nabla f(r,\Theta ,\phi )={\boldsymbol {\hat {r}}}{\frac {\partial }{\partial r}}+{\frac {1}{r\sin(\Theta )}}{\boldsymbol {\hat {\phi }}}{\frac {\partial }{\partial \phi }}+{\frac {1}{r}}{\boldsymbol {\hat {\Theta }}}{\frac {\partial }{\partial \Theta }}$

$\mathbf {A} (r,\Theta ,\phi )={\begin{pmatrix}A_{x}\\A_{y}\\A_{z}\end{pmatrix}}=A_{r}{\boldsymbol {\hat {r}}}+A_{\Theta }{\boldsymbol {\hat {\Theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}={\begin{pmatrix}A_{r}\sin(\Theta )\cos(\phi )+A_{\Theta }\cos(\Theta )\cos(\phi )-A_{\phi }\sin(\phi )\\A_{r}\sin(\Theta )\sin(\phi )+A_{\Theta }\cos(\Theta )\sin(\phi )+A_{\phi }\cos(\phi )\\A_{r}\cos(\Theta )-A_{\Theta }\sin(\Theta )\end{pmatrix}}$

${\begin{matrix}{\frac {\partial A_{x}}{\partial x}}&=&\left(\sin(\Theta )\cos(\phi ){\frac {\partial }{\partial r}}-{\frac {\sin(\phi )}{r\sin(\Theta )}}{\frac {\partial }{\partial \phi }}+{\frac {1}{r}}\cdot \cos(\Theta )\cos(\phi ){\frac {\partial }{\partial \Theta }}\right)\left(A_{r}\sin(\Theta )\cos(\phi )+A_{\Theta }\cos(\Theta )\cos(\phi )-A_{\phi }\sin(\phi )\right)\\&=&\sin ^{2}(\Theta )\cos ^{2}(\phi ){\frac {\partial A_{r}}{\partial r}}+\sin(\Theta )\cos(\Theta )\cos ^{2}(\phi ){\frac {\partial A_{\Theta }}{\partial r}}-\sin(\Theta )\cos(\phi )\sin(\phi ){\frac {\partial A_{\phi }}{\partial r}}\\&&-{\frac {\sin(\phi )\cos(\phi )}{r}}{\frac {\partial A_{r}}{\partial \phi }}+A_{r}{\frac {\sin ^{2}(\phi )}{r}}-{\frac {\sin(\phi )\cos(\phi )\cos(\Theta )}{r\sin(\Theta )}}{\frac {\partial A_{\Theta }}{\partial \phi }}+A_{\Theta }{\frac {\sin ^{2}(\phi )\cos(\Theta )}{r\sin(\Theta )}}+{\frac {\sin ^{2}(\phi )}{r\sin(\Theta )}}{\frac {\partial A_{\phi }}{\partial \phi }}+A_{\phi }{\frac {\sin(\phi )\cos(\phi )}{r\sin(\Theta )}}\\&&+{\frac {1}{r}}\cos(\Theta )\sin(\Theta )\cos ^{2}(\phi ){\frac {\partial A_{r}}{\partial \Theta }}+{\frac {1}{r}}A_{r}\cos ^{2}(\Theta )\cos ^{2}(\phi )\\&&+{\frac {1}{r}}\cos ^{2}(\Theta )\cos ^{2}(\phi ){\frac {\partial A_{\Theta }}{\partial \Theta }}-{\frac {1}{r}}A_{\Theta }\cos(\Theta )\sin(\Theta )\cos ^{2}(\phi )-{\frac {1}{r}}\cos(\Theta )\cos(\phi )\sin(\phi ){\frac {\partial A_{\phi }}{\partial \Theta }}\end{matrix}}$

${\begin{matrix}{\frac {\partial A_{y}}{\partial y}}&=&\left(\sin(\Theta )\sin(\phi ){\frac {\partial }{\partial r}}+{\frac {\cos(\phi )}{r\sin(\Theta )}}{\frac {\partial }{\partial \phi }}+{\frac {1}{r}}\cdot \cos(\Theta )\sin(\phi ){\frac {\partial }{\partial \Theta }}\right)\left(A_{r}\sin(\Theta )\sin(\phi )+A_{\Theta }\cos(\Theta )\sin(\phi )+A_{\phi }\cos(\phi )\right)\\&=&\sin ^{2}(\Theta )\sin ^{2}(\phi ){\frac {\partial A_{r}}{\partial r}}+\sin(\Theta )\cos(\Theta )\sin ^{2}(\phi ){\frac {\partial A_{\Theta }}{\partial r}}+\sin(\Theta )\cos(\phi )\sin(\phi ){\frac {\partial A_{\phi }}{\partial r}}\\&&+{\frac {\sin(\phi )\cos(\phi )}{r}}{\frac {\partial A_{r}}{\partial \phi }}+A_{r}{\frac {\cos ^{2}(\phi )}{r}}+{\frac {\sin(\phi )\cos(\phi )\cos(\Theta )}{r\sin(\Theta )}}{\frac {\partial A_{\Theta }}{\partial \phi }}+A_{\Theta }{\frac {\cos ^{2}(\phi )\cos(\Theta )}{r\sin(\Theta )}}+{\frac {\cos ^{2}(\phi )}{r\sin(\Theta )}}{\frac {\partial A_{\phi }}{\partial \phi }}-A_{\phi }{\frac {\sin(\phi )\cos(\phi )}{r\sin(\Theta )}}\\&&+{\frac {1}{r}}\cos(\Theta )\sin(\Theta )\sin ^{2}(\phi ){\frac {\partial A_{r}}{\partial \Theta }}+{\frac {1}{r}}A_{r}\cos ^{2}(\Theta )\sin ^{2}(\phi )\\&&+{\frac {1}{r}}\cos ^{2}(\Theta )\sin ^{2}(\phi ){\frac {\partial A_{\Theta }}{\partial \Theta }}-{\frac {1}{r}}A_{\Theta }\cos(\Theta )\sin(\Theta )\sin ^{2}(\phi )+{\frac {1}{r}}\cos(\Theta )\sin(\phi )\cos(\phi ){\frac {\partial A_{\phi }}{\partial \Theta }}\end{matrix}}$

${\begin{matrix}{\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}&=&\sin ^{2}(\Theta ){\frac {\partial A_{r}}{\partial r}}+\sin(\Theta )\cos(\Theta ){\frac {\partial A_{\Theta }}{\partial r}}+{\frac {A_{r}}{r}}+{\frac {A_{\Theta }\cos(\Theta )}{r\sin(\Theta )}}+{\frac {1}{r\sin(\Theta )}}{\frac {\partial A_{\phi }}{\partial \phi }}\\&&+{\frac {1}{r}}\sin(\Theta )\cos(\Theta ){\frac {\partial A_{r}}{\partial \Theta }}+{\frac {1}{r}}\cos ^{2}(\Theta )A_{r}+{\frac {1}{r}}\cos ^{2}(\Theta ){\frac {\partial A_{\Theta }}{\partial \Theta }}-{\frac {1}{r}}\sin(\Theta )\cos(\Theta )A_{\Theta }\end{matrix}}$

${\begin{matrix}{\frac {\partial A_{z}}{\partial z}}&=&\left(\cos(\Theta ){\frac {\partial }{\partial r}}-{\frac {1}{r}}\cdot \sin(\Theta ){\frac {\partial }{\partial \Theta }}\right)\left(A_{r}\cos(\Theta )-A_{\Theta }\sin(\Theta )\right)\\&=&\cos ^{2}{\Theta }{\frac {\partial A_{r}}{\partial r}}-\sin(\Theta )\cos(\Theta ){\frac {\partial A_{\Theta }}{\partial r}}-{\frac {1}{r}}\sin(\Theta )\cos(\Theta ){\frac {\partial A_{r}}{\partial \Theta }}+{\frac {1}{r}}A_{r}\sin ^{2}(\Theta )+{\frac {1}{r}}A_{\Theta }\sin(\Theta )\cos(\Theta )+{\frac {1}{r}}\sin ^{2}(\Theta ){\frac {\partial A_{\Theta }}{\partial \Theta }}\end{matrix}}$

$\nabla \cdot \mathbf {A} ={\frac {\partial A_{r}}{\partial r}}+{\frac {A_{r}}{r}}+{\frac {A_{\Theta }\cos(\Theta )}{r\sin(\Theta )}}+{\frac {1}{r\sin(\Theta )}}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {A_{r}}{r}}+{\frac {1}{r}}{\frac {\partial A_{\Theta }}{\partial \Theta }}={\frac {1}{r^{2}}}{\frac {\partial r^{2}A_{r}}{\partial r}}+{\frac {1}{r\sin(\Theta )}}{\frac {\partial }{\partial \Theta }}A_{\Theta }\sin(\Theta )+{\frac {1}{r\sin(\Theta )}}{\frac {\partial A_{\phi }}{\partial \phi }}$

$\Delta f(r,\Theta ,\phi )=\nabla \cdot (\nabla f(r,\Theta ,\phi ))=\nabla \cdot \left({\boldsymbol {\hat {r}}}{\frac {\partial }{\partial r}}+{\frac {1}{r\sin(\Theta )}}{\boldsymbol {\hat {\phi }}}{\frac {\partial }{\partial \phi }}+{\frac {1}{r}}{\boldsymbol {\hat {\Theta }}}{\frac {\partial }{\partial \Theta }}\right)f(r,\Theta ,\phi )$

$\Delta f(r,\Theta ,\phi )=\left({\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}r^{2}{\frac {\partial }{\partial r}}+{\frac {1}{r^{2}\sin(\Theta )}}{\frac {\partial }{\partial \Theta }}\sin(\Theta ){\frac {\partial }{\partial \Theta }}+{\frac {1}{r^{2}\sin(\Theta )^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}\right)f(r,\Theta ,\phi )$

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