ellipsoidal coordinates

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1: 23.21 Physical Applications

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§23.21(iii) Ellipsoidal Coordinates

►Ellipsoidal coordinates $(\xi ,\eta ,\zeta )$ may be defined as the three roots $\rho$ of the equation …The Laplacian operator ${\nabla }^2$ (§1.5(ii)) is given by ►

23.21.2 $$(\eta -\zeta )(\zeta -\xi )(\xi -\eta ){\nabla }^2=(\zeta -\eta )f(\xi )f^{\prime }(\xi )\frac{\partial }{\partial \xi }+(\xi -\zeta )f(\eta )f^{\prime }(\eta )\frac{\partial }{\partial \eta }+(\eta -\xi )f(\zeta )f^{\prime }(\zeta )\frac{\partial }{\partial \zeta },$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\xi$: ellipsoidal coordinate, $\eta$: ellipsoidal coordinate, $\zeta$: ellipsoidal coordinate and $f(\rho )$: function

Permalink:

http://dlmf.nist.gov/23.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.21(iii), §23.21 and Ch.23

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2: 29.18 Mathematical Applications

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§29.18(ii) Ellipsoidal Coordinates

►The wave equation (29.18.1), when transformed to ellipsoidal coordinates $\alpha ,\beta ,\gamma$: … ►

29.18.10 $$u(\alpha ,\beta ,\gamma )=u_1(\alpha )u_2(\beta )u_3(\gamma ),$$

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Symbols:

$\alpha$: ellipsoidal coordinate, $\beta$: ellipsoidal coordinate and $\gamma$: ellipsoidal coordinate

Permalink:

http://dlmf.nist.gov/29.18.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §29.18(ii), §29.18 and Ch.29

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3: 30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►The coordinate surfaces $\xi =\text{const}.$ are prolate ellipsoids of revolution with foci at $x=y=0$, $z=\pm c$. …

4: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►The coordinate surfaces $\xi =\text{const}.$ are oblate ellipsoids of revolution with focal circle $z=0$, $x^2+y^2=c^2$. …