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Regel 136:
=== Cilindercoördinaten ===
==== Gradiënt van een scalaire functie ====
:<math>\nabla f = (\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})\,</math>
 
De component van grad f in de richting van <math>\rho</math> is:
:<math>(\nabla f)_\rho =
\frac{\partial f}{\partial x}\cos(\phi)+
\frac{\partial f}{\partial y}\sin(\phi)= \frac{\partial f_c}{\partial \rho}\,</math>
 
De component van grad f in de richting van <math>\phi</math> is:
Regel 147:
\frac{\partial f}{\partial x}\sin(\phi)+
\frac{\partial f}{\partial y}\cos(\phi)=\frac 1\rho\frac{\partial f_c}{\partial \phi}
\,
</math>
 
==== Divergentie van een vectorveld ====
 
:<math>\nabla\cdot A= \frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\,</math>
 
We drukken de divergentie uit in de componenten van de polaire voorstelling:
:<math>A_x =A_\rho\cos(\phi)-A_\phi \sin(\phi)\,</math>
:<math>A_y =A_\rho\sin(\phi)+A_\phi \cos(\phi)\,</math>
 
dus
Regel 170 ⟶ 169:
\frac{\partial}{\partial \phi}\left(A_\rho\cos(\phi)-A_\phi\sin(\phi)\right)\frac{\partial \phi}{\partial x}
=
\,</math>
 
::<math>
\frac{\partial}{\partial \rho}\left(A_\rho\cos(\phi)-A_\phi\sin(\phi)\right)\cos(\phi)+
\frac{\partial}{\partial \phi}\left(A_\rho\cos(\phi)-A_\phi\sin(\phi)\right)(-\frac 1\rho\sin(\phi))=
\,</math>
 
::<math>
Regel 184 ⟶ 183:
\frac 1\rho \frac{\partial A_\phi}{\partial \phi}\sin^2(\phi)+
\frac 1\rho A_\phi \sin(\phi)\cos(\phi)
\,</math>
 
 
Regel 196 ⟶ 195:
\frac{\partial}{\partial \phi}\left(A_\rho\sin(\phi)+A_\phi\cos(\phi)\right) \frac{\partial \phi}{\partial y}
=
\,</math>
 
::<math>
Regel 202 ⟶ 201:
\frac{\partial}{\partial \phi}\left(A_\rho\sin(\phi)+A_\phi\cos(\phi)\right)(\frac 1\rho\cos(\phi))
=
\,</math>
::<math>
\frac{\partial A_\rho}{\partial \rho}\sin^2(\phi)+
Regel 210 ⟶ 209:
\frac 1\rho \frac{\partial A_\phi}{\partial \phi}\cos^2(\phi)-
\frac 1\rho A_\phi \cos(\phi)\sin(\phi)
\,</math>
 
Samen leidt dat tot:
Regel 219 ⟶ 218:
\frac{A_\rho}{\rho}+
\frac 1{\rho}\frac{\partial A_\phi}{\partial \phi}+
\frac{\partial A_z}{\partial z}\,</math>
 
=== Rotatie van een vectorveld ===
Regel 234 ⟶ 233:
-\frac{\partial A_y}{\partial z}
=
\,</math>
:::<math>
\frac{\partial A_z}{\partial \rho}\sin(\phi)
Regel 240 ⟶ 239:
-\frac{\partial A_\rho}{\partial z}\sin(\phi)
-\frac{\partial A_\phi}{\partial z}\cos(\phi)
\,</math>
 
<!--y-->
Regel 253 ⟶ 252:
-\frac{\partial A_z}{\partial \phi}\frac{\partial \phi}{\partial x}
=
\,</math>
:::<math>
\frac{\partial A_\rho}{\partial z}\cos(\phi)-\frac{\partial A_\phi}{\partial z}\sin(\phi)
-\frac{\partial A_z}{\partial \rho}\cos(\phi)
+\frac{\partial A_z}{\partial \phi}\frac 1\rho\sin(\phi)
\,</math>
 
<!--z-->
Regel 273 ⟶ 272:
-\frac{\partial A_x}{\partial \phi}\frac{\partial \phi}{\partial y}
=
\,</math>
 
<br />
:::<math>
\frac{\partial (A_\rho\sin(\phi)+A_\phi \cos(\phi))}{\partial \rho}\cos(\phi)
-\frac{\partial (A_\rho\sin(\phi)+A_\phi \cos(\phi))}{\partial \phi}\frac 1\rho\sin(\phi)
\,</math>
 
:::<math>
Regel 284 ⟶ 283:
-\frac{\partial (A_\rho\cos(\phi)-A_\phi \sin(\phi))}{\partial \phi}\frac 1\rho\cos(\phi)
=
\,</math>
 
<br />
:::<math>
\frac{\partial A_\rho}{\partial \rho}\sin(\phi)\cos(\phi)+\frac{\partial A_\phi}{\partial \rho}\cos^2(\phi)
-\frac{\partial A_\rho\sin(\phi)}{\partial \phi}\frac 1\rho\sin(\phi)
-\frac{\partial A_\phi \cos(\phi)}{\partial \phi}\frac 1\rho\sin(\phi)
\,</math>
 
:::<math>
Regel 298 ⟶ 297:
+\frac{\partial A_\phi \sin(\phi))}{\partial \phi}\frac 1\rho\cos(\phi)
=
\,</math>
 
<br />
:::<math>
\frac{\partial A_\phi}{\partial \rho}
-\frac{\partial A_\rho}{\partial \phi}\frac 1\rho
+A_\phi \frac 1\rho
\,</math>
 
Transformeren naar ρ en φ:
Regel 317 ⟶ 316:
\frac 1\rho\frac{\partial A_z}{\partial \phi}
-\frac{\partial A_\phi}{\partial z}
\,</math>
 
<!--phi-->
Regel 328 ⟶ 327:
\frac{\partial A_\rho}{\partial z}
-\frac{\partial A_z}{\partial \rho}
</math>
 
\,</math>
 
==== Laplaciaan van een scalaire functie ====
Uit het bovenstaande volgt voor de Laplaciaanlaplaciaan van een scalaire functie ''<math>f''</math>:
:<math>\Delta f = \nabla\cdot\nabla f =
\frac{\partial (\nabla f)_\rho}{\partial \rho}+
\frac{(\nabla f)_\rho}{\rho}+
\frac 1{\rho}\frac{\partial (\nabla f)_\phi}{\partial \phi}+
\frac{\partial (\nabla f)_z}{\partial z}=\,
</math>
::<math>
\frac{\partial \frac{\partial f_c}{\partial \rho}}{\partial \rho}+
Regel 346 ⟶ 345:
\frac{1}{\rho}\frac{\partial f_c}{\partial \rho}+
\frac 1{\rho^2}\frac{\partial^2 f_c}{\partial \phi^2}+
\frac{\partial^2 f_c}{\partial z^2}\,</math>
 
==== Laplaciaan van een vectorveld ====
Uit het bovenstaande volgt voor de Laplaciaanlaplaciaan van een vectorveld ''<math>A''</math>:
:<math>(\Delta A)_\rho = (\Delta A)_x\cos(\phi)+(\Delta A)_y\sin(\phi)=\Delta A_x\cos(\phi)+\Delta A_y\sin(\phi)=
\,</math>
 
::<math>
Regel 359 ⟶ 358:
\frac{\partial^2 A_x}{\partial z^2}\right)\cos(\phi)
+
\,</math>
 
::<math>
Regel 365 ⟶ 364:
\frac{1}{\rho}\frac{\partial A_y}{\partial \rho}+
\frac 1{\rho^2}\frac{\partial^2 A_y}{\partial \phi^2}+
\frac{\partial^2 A_y}{\partial z^2}\right)\sin(\phi)=\,
</math>
 
::<math>
Regel 376:
\right)+
\frac{\partial^2 A_\rho}{\partial z^2}=
\,</math>
 
::<math>
Regel 383:
\frac{\partial^2 A_\rho}{\partial z^2}
+
\,</math>
 
::<math>
Regel 391:
\frac{\partial^2 (A_\rho\sin(\phi)+A_\phi \cos(\phi))}{\partial \phi^2}\sin(\phi)
\right)
=
=\,</math>
</math>
 
::<math>
Regel 397 ⟶ 398:
\frac{1}{\rho}\frac{\partial A_\rho}{\partial \rho}+
\frac{\partial^2 A_\rho}{\partial z^2}
+\frac 1{\rho^2}\frac{\partial^2 A_\rho}{\partial \phi^2}
+
-\frac 1{\rho^2}\frac{\partial^2 A_\rho}{\partial \phi^2}
-\frac 2{\rho^2}\frac{\partial A_\phi}{\partial \phi}
-
=
\frac 1{\rho^2}A_\rho
</math>
-
\frac 2{\rho^2}\frac{\partial A_\phi}{\partial \phi}
=\,</math>
 
::<math>
\Delta A_\rho
-\frac 1{\rho^2}A_\rho
-
-\frac 12{\rho^2}\frac{\partial A_\rhophi}{\partial \phi}
</math>
-
\frac 2{\rho^2}\frac{\partial A_\phi}{\partial \phi}
\,</math>
 
En analoog:
 
:<math>
:<math>(\Delta A)_\phi = -(\Delta A)_x\sin(\phi)+(\Delta A)_y\cos(\phi)=-\Delta A_x\sin(\phi)+\Delta A_y\cos(\phi)=
(\Delta A)_\phi =
\,</math>
-(\Delta A)_x\sin(\phi)+(\Delta A)_y\cos(\phi)=
-\Delta A_x\sin(\phi)+\Delta A_y\cos(\phi)=
</math>
 
::<math>
Regel 424:
\frac{\partial^2 A_x}{\partial z^2}\right)\sin(\phi)
+
\,</math>
 
::<math>
Regel 430:
\frac{1}{\rho}\frac{\partial A_y}{\partial \rho}+
\frac 1{\rho^2}\frac{\partial^2 A_y}{\partial \phi^2}+
\frac{\partial^2 A_y}{\partial z^2}\right)\cos(\phi)=\,
</math>
 
::<math>
Regel 441 ⟶ 442:
\right)+
\frac{\partial^2 A_\phi}{\partial z^2}=
\,</math>
 
::<math>
Regel 447 ⟶ 448:
\frac{1}{\rho}\frac{\partial A_\phi}{\partial \rho}+
\frac{\partial^2 A_\phi}{\partial z^2}-
\,</math>
 
::<math>
Regel 454 ⟶ 455:
\frac{\partial^2 (A_\rho\cos(\phi)-A_\phi \sin(\phi))}{\partial \phi^2}\sin(\phi)-
\frac{\partial^2 (A_\rho\sin(\phi)+A_\phi \cos(\phi))}{\partial \phi^2}\cos(\phi)
 
\right)
=
=\,</math>
</math>
 
::<math>
Regel 466:
\frac 1{\rho^2}A_\phi+
\frac 2{\rho^2}\frac{\partial A_\rho}{\partial \phi}
=
=\,</math>
</math>
 
::<math>
\Delta A_\phi
-\frac 1{\rho^2}A_\phi
-
+\frac 12{\rho^2}\frac{\partial A_\rho}{\partial \phi}
</math>
+
\frac 2{\rho^2}\frac{\partial A_\rho}{\partial \phi}
\,</math>
 
== Bolcoördinaten ==
==== Gradiënt van een scalaire functie ====
:<math>
:<math>\nabla f = (\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})\,</math>
\nabla f = (\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})
</math>
 
De component van <math>\operatorname{grad} f</math> in de richting van ''<math>r''</math> is:
:<math>(\nabla f)_r =
\frac{\partial f}{\partial x}\cos(\phi)\sin(\theta)+
Regel 491 ⟶ 492:
=
\frac{\partial f_b}{\partial r}
\,</math>
 
De component van <math>\operatorname{grad} f</math> in de richting van <math>\phi</math> is:
:<math>(\nabla f)_\phi = -
\frac{\partial f}{\partial x}\sin(\phi)+
Regel 504 ⟶ 505:
=
\frac 1{r\sin(\theta)}\frac{\partial f_b}{\partial \phi}
\,</math>
 
De component van <math>\operatorname{grad} f</math> in de richting van <math>\theta</math> is:
:<math>(\nabla f)_\theta =
\frac{\partial f}{\partial x}\cos(\phi)\cos(\theta)+
Regel 519 ⟶ 520:
=
\frac 1r\frac{\partial f_b}{\partial \theta}
\,</math>
 
==== Divergentie van een vectorveld ====
:<math>\nabla\cdot A= \frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}</math>
 
:<math>\nabla\cdot A= \frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\,</math>
 
We drukken de divergentie uit in de voorstelling in bolcoördinaten:
:<math>A_x =A_r\cos(\phi)\sin(\theta)-A_\phi\sin(\phi)+A_\theta\cos(\phi)\cos(\theta)\,</math>
:<math>A_y =A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta)\,</math>
:<math>A_z =A_r\cos(\theta)-A_\theta\sin(\theta)\,</math>
 
Nu is:
Regel 539:
</math>
 
en analoog voor <math>y</math> en <math>z</math>, zodat:
 
:<math>\nabla\cdot A=\,</math>
<!--x-->
::<math>
Regel 549:
\right)
\cos(\phi)\sin(\theta)+
\,</math>
 
::<math>
Regel 557:
\right)
(-\frac 1{r\sin(\theta)})\sin(\phi)+
\,</math>
 
::<math>
Regel 566:
\frac 1{r}\cos(\phi)\cos(\theta)
+
\,</math>
 
 
Regel 576:
\right)
\sin(\phi)\sin(\theta)+
\,</math>
 
::<math>
Regel 584:
\right)
\frac 1{r\sin(\theta)}\cos(\phi)+
\,</math>
 
::<math>
Regel 593:
\frac 1{r}\sin(\phi)\cos(\theta)
+
\,</math>
 
 
Regel 603:
\right)
\cos(\theta)+
\,</math>
 
::<math>
Regel 611:
\right)
(-\frac 1r)\sin(\theta)
\,</math>
 
We verzamelen apart:
Regel 627:
\right)
=\frac 2{r}A_r
\,</math>
 
:<math>
Regel 633:
\left(
\cos^2(\phi)\sin^2(\theta)
+\sin^2(\phi)\sin^2(\theta)
+
+\sin^2(\phi)\sincos^2(\theta)
+
\cos^2(\theta)
\right)
=
\frac{\partial A_r}{\partial r}
\,</math>
 
:<math>
Regel 646 ⟶ 644:
\left(
-\sin(\phi)\cos(\phi)\sin(\theta)
+\cos(\phi)\sin(\phi)\sin(\theta)
+
\cos(\phi)\sin(\phi)\sin(\theta)
\right)
=0
\,</math>
 
:<math>
Regel 656 ⟶ 653:
\left(
\cos(\phi)\sin(\theta)(-\frac 1{r})\cos(\phi)\cos(\theta)
+\sin(\phi)\sin(\theta)\frac 1{r}\sin(\phi)\cos(\theta)
+
+\sin(\phi)\sincos(\theta)\frac 1{r}\sin(\phi)\cos(\theta)
+
\cos(\theta)\frac 1{r}\sin(\theta)
 
\right)
=0
\,</math>
 
de termen waarin <math>A_\phi</math> voorkomt:
Regel 674 ⟶ 668:
\right)
=0
\,</math>
 
:<math>
Regel 680 ⟶ 674:
\left(
-\sin(\phi)\cos(\phi)\sin(\theta)
+\cos(\phi)\sin(\phi)\sin(\theta)
+
\cos(\phi)\sin(\phi)\sin(\theta)
\right)
=0
\,</math>
 
:<math>
Regel 690 ⟶ 683:
\left(
\sin(\phi)\frac 1{r\sin(\theta)}\sin(\phi)
+\cos(\phi)\frac 1{r\sin(\theta)}\cos(\phi)
+
\cos(\phi)\frac 1{r\sin(\theta)}\cos(\phi)
\right)
=
\frac 1{r\sin(\theta)}\frac{\partial A_\phi}{\partial \phi}
\,</math>
 
:<math>
Regel 701 ⟶ 693:
\left(
-\frac 1{r}\sin(\phi)\cos(\phi)\cos(\theta)
+\frac 1{r}\cos(\phi)\sin(\phi)\cos(\theta)
+
\frac 1{r}\cos(\phi)\sin(\phi)\cos(\theta)
\right)
=0
\,</math>
 
 
Regel 719 ⟶ 710:
\frac 1{r}\sin(\theta)\cos(\theta)
\right)
\,</math>
 
::<math>=\frac 1{r\sin(\theta)}A_\theta\cos(\theta)</math>
::<math>
=\frac 1{r\sin(\theta)}A_\theta\cos(\theta)
\,</math>
 
:<math>
Regel 729 ⟶ 718:
\left(
\cos^2(\phi)\cos(\theta)\sin(\theta)
+\sin^2(\phi)\cos(\theta)\sin(\theta)
+
-\sin^2(\phitheta)\cos(\theta)\sin(\theta)
-
\sin(\theta)\cos(\theta)
\right)
=0
\,</math>
 
:<math>
Regel 741 ⟶ 728:
\left(
-\frac 1{r\sin(\theta)}\cos(\phi)\sin(\phi)\cos(\theta)
+\frac 1{r\sin(\theta)}\sin(\phi)\cos(\phi)\cos(\theta)
+
\frac 1{r\sin(\theta)}\sin(\phi)\cos(\phi)\cos(\theta)
\right)
=
0
\,</math>
 
:<math>
Regel 752 ⟶ 738:
\left(
\frac 1r\cos^2(\phi)\cos^2(\theta)
+\frac 1r\sin^2(\phi)\cos^2(\theta)
+
+\frac 1r\sin^2(\phi)\cos^2(\theta)
+
 
\frac 1r\sin^2(\theta)
 
\right)
=
\frac{\partial A_\theta}{\partial \theta}
\,</math>
 
Samen geeft dat:
Regel 767 ⟶ 749:
:<math>\nabla\cdot A=
\frac 2{r}A_r
+\frac{\partial A_r}{\partial r}
+
+\frac 1{r\sin(\theta)}\frac{\partial A_rA_\phi}{\partial r\phi}
+\frac 1{r\sin(\theta)}A_\theta\cos(\theta)
+
\frac 1{r\sin(\theta)}+\frac{\partial A_\phitheta}{\partial \phitheta}
+
\frac 1{r\sin(\theta)}A_\theta\cos(\theta)
+
\frac{\partial A_\theta}{\partial \theta}
=
\,</math>
 
::<math>
\frac 1{r^2}\frac{\partial r^2A_r}{\partial r}
+\frac 1{r\sin(\theta)}\frac{\partial A_\phi}{\partial \phi}
+
+\frac 1{r\sin(\theta)}\frac{\partial \sin(\theta)A_\phitheta}{\partial \phitheta}
</math>
+
\frac 1{r\sin(\theta)}\frac{\partial \sin(\theta)A_\theta}{\partial \theta}
\,</math>
 
 
Regel 790 ⟶ 766:
Voor de divergentie bepaalden we <math>
\frac{\partial A_x}{\partial x}, \frac{\partial A_y}{\partial y}, \frac{\partial A_z}{\partial z}
\,</math>. Nu moeten de andere afgeleiden bepaald worden.
 
<!--x-->
Regel 804 ⟶ 780:
+\frac{\partial A_z}{\partial \theta}\frac{\partial \theta}{\partial y}
=
\,</math>
 
::<math>
Regel 812 ⟶ 788:
\right)
\sin(\phi)\sin(\theta)+
\,</math>
 
::<math>
Regel 820 ⟶ 796:
\right)
\frac 1{r\sin(\theta)}\cos(\phi)+
\,</math>
 
::<math>
Regel 828 ⟶ 804:
\right)
\frac 1{r}\sin(\phi)\cos(\theta)
\,</math>
 
 
Regel 839 ⟶ 815:
+\frac{\partial A_y}{\partial \theta}\frac{\partial \theta}{\partial z}
=
\,</math>
 
::<math>
Regel 847 ⟶ 823:
\right)
\cos(\theta)+
\,</math>
 
::<math>
Regel 855 ⟶ 831:
\right)
(-\frac 1r)\sin(\theta)
\,</math>
 
<!--rotx-->
Regel 865 ⟶ 841:
\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}
=
\,</math>
 
::<math>
\frac{\partial A_r}{\partial \phi}
\frac 1{r\sin(\theta)}\cos(\phi)\cos(\theta)
+\frac{\partial A_r}{\partial \theta}\frac 1{r}\sin(\phi)
 
</math>
+\frac{\partial A_r}{\partial \theta}
\frac 1{r}\sin(\phi)
\,</math>
 
::<math>
-\frac{\partial A_\phi}{\partial r}
\cos(\phi)\cos(\theta)
 
+\frac{\partial A_\phi}{\partial \theta}
\frac 1r\cos(\phi)\sin(\theta)
\,</math>
 
::<math>
-\frac{\partial A_\theta}{\partial r}\sin(\phi)
-\frac{\partial A_\theta}{\partial \phi}\frac 1{r}\cos(\phi)
\sin(\phi)
-A_\theta\frac 1{r}\sin(\phi)
 
</math>
-\frac{\partial A_\theta}{\partial \phi}
\frac 1{r}\cos(\phi)
 
-A_\theta
\frac 1{r}\sin(\phi)
\,</math>
 
<!--y-->
Regel 906 ⟶ 874:
+\frac{\partial A_x}{\partial \theta}\frac{\partial \theta}{\partial z}
=
\,</math>
 
::<math>
Regel 914 ⟶ 882:
\right)
\cos(\theta)+
\,</math>
 
::<math>
Regel 922 ⟶ 890:
\right)
(-\frac 1r)\sin(\theta)
\,</math>
 
 
Regel 933 ⟶ 901:
+\frac{\partial A_z}{\partial \theta}\frac{\partial \theta}{\partial x}
=
\,</math>
 
::<math>
Regel 941 ⟶ 909:
\right)
\cos(\phi)\sin(\theta)+
\,</math>
 
::<math>
Regel 949 ⟶ 917:
\right)
(-\frac 1{r\sin(\theta)})\sin(\phi)+
\,</math>
 
::<math>
Regel 957 ⟶ 925:
\right)
\frac 1{r}\cos(\phi)\cos(\theta)
\,</math>
 
<!--rotyrot y-->
 
Daaruit volgt:
Regel 967 ⟶ 935:
\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}
=
\,</math>
 
::<math>
Regel 974 ⟶ 942:
-\frac{\partial A_r}{\partial \theta}
\frac 1r\cos(\phi)
\,</math>
 
::<math>
-\frac{\partial A_\phi}{\partial r} \sin(\phi)\cos(\theta)
-
+\frac{\partial A_\phi}{\partial r\theta} \frac 1r\sin(\phi)\sin(\theta)
</math>
\sin(\phi)\cos(\theta)
+
\frac{\partial A_\phi}{\partial \theta}
\frac 1r\sin(\phi)\sin(\theta)
\,</math>
 
::<math>
+A_\theta \frac 1r\cos(\phi)
+
+\frac{\partial A_\theta}{\partial r} \cos(\phi)
A_\theta
-\frac{\partial A_\theta}{\partial \phi} \frac 1r1{r}\cossin(\phi)
</math>
+
\frac{\partial A_\theta}{\partial r}
\cos(\phi)
-
\frac{\partial A_\theta}{\partial \phi}
\frac 1{r}\sin(\phi)
\,</math>
 
<!--z-->
Regel 1.009 ⟶ 967:
+\frac{\partial A_x}{\partial \theta}\frac{\partial \theta}{\partial y}
=
\,</math>
 
::<math>
Regel 1.017 ⟶ 975:
\right)
\sin(\phi)\sin(\theta)+
\,</math>
 
::<math>
Regel 1.025 ⟶ 983:
\right)
\frac 1{r\sin(\theta)}\cos(\phi)+
\,</math>
 
::<math>
Regel 1.033 ⟶ 991:
\right)
\frac 1{r}\sin(\phi)\cos(\theta)
\,</math>
 
<!--y/x-->
Regel 1.043 ⟶ 1.001:
+\frac{\partial A_y}{\partial \theta}\frac{\partial \theta}{\partial x}
=
\,</math>
 
::<math>
Regel 1.051 ⟶ 1.009:
\right)
\cos(\phi)\sin(\theta)+
\,</math>
 
::<math>
Regel 1.059 ⟶ 1.017:
\right)
(-\frac 1{r\sin(\theta)})\sin(\phi)+
\,</math>
 
::<math>
Regel 1.067 ⟶ 1.025:
\right)
\frac 1{r}\cos(\phi)\cos(\theta)
\,</math>
 
 
<!--rotzrot z-->
 
Daaruit volgt:
Regel 1.079 ⟶ 1.037:
-\frac{\partial A_x}{\partial y}
=
\,</math>
 
::<math>
\frac{\partial A_r}{\partial \phi}
\frac 1{r}
-A_\phi \frac 1{r\sin(\theta)}
-
-\frac{\partial A_\phi}{\partial r} \sin(\theta)
A_\phi
-\frac{\partial A_\phi}{\partial \theta} \frac 1{r}\sincos(\theta)}
+\frac{\partial A_\theta}{\partial \phi} \frac 1{r\sin(\theta)}\cos(\theta)
-
</math>
\frac{\partial A_\phi}{\partial r}
\sin(\theta)
-
\frac{\partial A_\phi}{\partial \theta}
\frac 1{r}\cos(\theta)
+
\frac{\partial A_\theta}{\partial \phi}
\frac 1{r\sin(\theta)}\cos(\theta)
\,</math>
 
 
Transformeren naar <math>r, φ\,\phi</math> en θ<math>\theta</math>:
 
<!--rho-->
Regel 1.110 ⟶ 1.060:
+(\nabla\times A)_z\cos(\theta)
=
\,</math>
 
::<math>
A_\phi \frac 1{r\sin(\theta)}\cos(\theta)
A_\phi
+\frac 11r \frac{r\sin(partial A_\theta)phi}{\cos(partial \theta)}
-\frac{\partial A_\theta}{\partial \phi} \frac 1{r\sin(\theta)}
+
</math>
\frac 1r
\frac{\partial A_\phi}{\partial \theta}
-
\frac{\partial A_\theta}{\partial \phi}
\frac 1{r\sin(\theta)}
\,</math>
 
 
Regel 1.131 ⟶ 1.076:
+(\nabla\times A)_y\cos(\phi)
=
\,</math>
::<math>
\frac{\partial A_\theta}{\partial r}
-\frac 1r \frac{\partial A_r}{\partial \theta}
-
+\frac 1r A_\theta
</math>
\frac{\partial A_r}{\partial \theta}
+
\frac 1r
A_\theta
\,</math>
 
 
Regel 1.151 ⟶ 1.092:
-(\nabla\times A)_z\sin(\theta)
=
\,</math>
 
::<math>
\frac 1r A_\phi
-\frac 1{r\sin(\theta)} \frac{\partial A_r}{\partial \phi}
A_\phi
+\frac{\partial A_\phi}{\partial r}
-
</math>
\frac 1{r\sin(\theta)}
\frac{\partial A_r}{\partial \phi}
+
\frac{\partial A_\phi}{\partial r}
\,</math>
 
==== Laplaciaan van een scalaire functie ====
Uit het bovenstaande volgt voor de Laplaciaanlaplaciaan van een scalaire functie ''<math>f''</math>:
 
:<math>
\Delta f = \nabla\cdot\nabla f =
 
\frac 1{r^2}\frac{\partial (r^2\frac{\partial f_b}{\partial r})}{\partial r}
+\frac 1{r\sin(\theta)}\frac{\partial (\frac 1{r\sin(\theta)}\frac{\partial f_b}{\partial \phi})}{\partial \phi}
+
+\frac 1{r\sin(\theta)}\frac{\partial (\frac 1{r\sin(\theta)}\frac 1r\frac{\partial f_b}{\partial \phitheta})}{\partial \phitheta}
+
\frac 1{r\sin(\theta)}\frac{\partial (\sin(\theta)\frac 1r\frac{\partial f_b}{\partial \theta})}{\partial \theta}
=
\,</math>
 
::<math>
\frac 1{r^2}\frac{\partial (r^2\frac{\partial f_b}{\partial r})}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 f_b}{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 (\sin(\theta)\frac{\partial f_b}{\partial \phi^2theta})}{\partial \theta}
</math>
+
\frac 1{r^2\sin(\theta)}\frac{\partial (\sin(\theta)\frac{\partial f_b}{\partial \theta})}{\partial \theta}
\,</math>
 
==== Laplaciaan van een vectorveld ====
Uit het bovenstaande volgt in bolcoördinaten voor de Laplaciaanlaplaciaan van een vectorveld ''<math>A''</math>:
 
:<math>
Regel 1.195 ⟶ 1.127:
+(\Delta A)_z\cos(\theta)
=
\,</math>
 
 
::<math>
Regel 1.203 ⟶ 1.134:
+\Delta A_z\cos(\theta)
=
\,</math>
 
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(A_x) \cos(\phi)\sin(\theta)
(A_x)
\cos(\phi)\sin(\theta)
+
\,</math>
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(A_y) \sin(\phi)\sin(\theta)
(A_y)
\sin(\phi)\sin(\theta)
+
\,</math>
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(A_z) \cos(\theta)
\cos(\theta)
=
\,</math>
 
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(A_r\cos(\phi)\sin(\theta)-A_\phi\sin(\phi)+A_\theta\cos(\phi)\cos(\theta)) \cos(\phi)\sin(\theta)
\cos(\phi)\sin(\theta)
+
\,</math>
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta)) \sin(\phi)\sin(\theta)
\sin(\phi)\sin(\theta)
+
\,</math>
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(A_r\cos(\theta)-A_\theta\sin(\theta)) \cos(\theta)
\cos(\theta)
=
\,</math>
 
 
::<math>
\frac 1{r^2}\frac{\partial r^2\frac{\partial A_r}{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 A_r}{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial A_r}{\partial \phi^2theta}}{\partial \theta}
-\frac {2}{r^2}A_r
+
-\frac 12{r^2\sin(\theta)}\frac{\partial A_\sin(\theta)\frac{\partial A_rphi}{\partial \theta}}{\partial \thetaphi}
-\frac {2\cos(\theta)}{r^2\sin(\theta)}A_\theta
-
-\frac {2}{r^2}A_r\frac{\partial A_\theta}{\partial \theta}
-
\frac 2{r^2\sin(\theta)}\frac{\partial A_\phi}{\partial \phi}
-
\frac {2\cos(\theta)}{r^2\sin(\theta)}A_\theta
-
\frac 2{r^2}\frac{\partial A_\theta}{\partial \theta}
=
\,</math>
 
 
::<math>
Regel 1.309 ⟶ 1.212:
-\frac{2}{r^2}\frac{\partial A_\theta}{\partial \theta}
-\frac{2}{r^2\sin(\theta)}\frac{\partial A_\phi}{\partial \phi}
\,</math>
 
En analoog voor de φ-component:
 
En analoog voor de <math>\phi</math>-component:
 
:<math>
Regel 1.319 ⟶ 1.221:
+(\Delta A)_y\cos(\phi)=
-\Delta A_x\sin(\phi)+\Delta A_y\cos(\phi)=
\,</math>
 
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(-A_x) \sin(\phi)
\sin(\phi)
+
\,</math>
 
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(A_y) \cos(\phi)
\cos(\phi)
=
\,</math>
 
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(-A_r\cos(\phi)\sin(\theta)+A_\phi\sin(\phi)-A_\theta\cos(\phi)\cos(\theta)) \sin(\phi)
\sin(\phi)
+
\,</math>
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2}{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta)) \cos(\phi)
\cos(\phi)
=
\,</math>
 
 
::<math>
\frac 1{r^2}\frac{\partial r^2\frac{\partial A_\phi}{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 A_\phi}{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial A_\phi}{\partial \phi^2theta}}{\partial \theta}
-\frac{A_\phi}{(r\sin(\theta))^2}
+
+\frac 1{r^2\sin(\theta)}\frac{\partial (r\sin(\theta))^2}\frac{\partial A_\phiA_r}{\partial \theta}}{\partial \thetaphi}
+\frac{2 \cos(\theta)}{(r\sin(\theta))^2}\frac{\partial A_\theta }{\partial \phi}
-
\frac{A_\phi}{(r\sin(\theta))^2}
+
\frac{2}{(r\sin(\theta))^2}\frac{\partial A_r}{\partial \phi}
+
\frac{2 \cos(\theta)}{(r\sin(\theta))^2}\frac{\partial A_\theta }{\partial \phi}
=
\,</math>
 
 
::<math>
\Delta A_\phi
-\frac{A_\phi}{(r\sin(\theta))^2}
-
+\frac{A_\phi2}{(r\sin(\theta))^2}\frac{\partial A_r}{\partial \phi}
+\frac{2 \cos(\theta)}{(r\sin(\theta))^2}\frac{\partial A_\theta}{\partial \phi}
+
</math>
\frac{2}{(r\sin(\theta))^2}\frac{\partial A_r}{\partial \phi}
+
\frac{2 \cos(\theta)}{(r\sin(\theta))^2}\frac{\partial A_\theta}{\partial \phi}
\,</math>
 
 
En analoog voor de θ<math>\theta</math>-component:
 
:<math>
Regel 1.413 ⟶ 1.290:
-(\Delta A)_z\sin(\theta)
=
\,</math>
 
 
::<math>
Regel 1.421 ⟶ 1.297:
-\Delta A_z\sin(\theta)
=
\,</math>
 
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(A_x) \cos(\phi)\cos(\theta)
(A_x)
\cos(\phi)\cos(\theta)
+
\,</math>
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(A_y) \sin(\phi)\cos(\theta)
(A_y)
\sin(\phi)\cos(\theta)
+
\,</math>
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(-A_z) \sin(\theta)
\sin(\theta)
=
\,</math>
 
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(A_r\cos(\phi)\sin(\theta)-A_\phi\sin(\phi)+A_\theta\cos(\phi)\cos(\theta)) \cos(\phi)\cos(\theta)
\cos(\phi)\cos(\theta)
+
\,</math>
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(A_r\sin(\phi)\sin(\theta)+A_\phi\cos(\phi)+A_\theta\sin(\phi)\cos(\theta)) \sin(\phi)\cos(\theta)
\sin(\phi)\cos(\theta)
+
\,</math>
 
::<math>
\left(
\frac 1{r^2}\frac{\partial r^2\frac{\partial }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial }{\partial \phi^2theta}}{\partial \theta}
+
\frac 1{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial }{\partial \theta}}{\partial \theta}
\right)
(-A_r\cos(\theta)+A_\theta\sin(\theta)) \sin(\theta)
\sin(\theta)
=
\,</math>
 
 
::<math>
\frac 1{r^2}\frac{\partial r^2\frac{\partial A_\theta }{\partial r}}{\partial r}
+\frac 1{(r\sin(\theta))^2}\frac{\partial^2 A_\theta }{\partial \phi^2}
+
+\frac 1{(r^2\sin(\theta))^2}\frac{\partial^2 \sin(\theta)\frac{\partial A_\theta }{\partial \phi^2theta}}{\partial \theta}
-\frac{A_\theta}{(r\sin(\theta))^2}
+
+\frac 1{2}{r^2\sin(\theta)}\frac{\partial \sin(\theta)\frac{\partial A_\theta }{\partial \theta}A_r}{\partial \theta}
-\frac{2 \cos(\theta)}{(r\sin(\theta))^2}\frac{\partial A_\phi}{\partial \phi}
-
\frac{A_\theta}{(r\sin(\theta))^2}
+
\frac{2}{r^2}\frac{\partial A_r}{\partial \theta}
-
\frac{2 \cos(\theta)}{(r\sin(\theta))^2}\frac{\partial A_\phi}{\partial \phi}
=
\,</math>
 
 
::<math>
\Delta A_\theta
-\frac{A_\theta}{(r\sin(\theta))^2}
-
+\frac{2}{r^2}\frac{\partial A_r}{\partial \theta}
\frac{A_\theta}{(r\sin(\theta))^2}
-\frac{2 \cos(\theta)}{(r\sin(\theta))^2}\frac{\partial A_\phi}{\partial \phi}
+
\frac{2}{r^2}\frac{\partial A_r}{\partial \theta}
-
\frac{2 \cos(\theta)}{(r\sin(\theta))^2}\frac{\partial A_\phi}{\partial \phi}
</math>
 
Overgenomen van "https://nl.wikipedia.org/wiki/Nabla_in_verschillende_assenstelsels"