Rotationally invariant Green's functions for the three-variable Laplace equation in all known coordinate...
$begingroup$
Green's function for the three-variable Laplace equation in Cartesian coordinates is
$$frac{1}{|mathbf{r}-mathbf{r'}|} = frac{1}{sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}$$
It may be written in spherical coordinates $(r, theta, phi)$ in Laplace expansion (see here or Section 3.9 of Jackson book "Classical Electrodynamics") using spherical harmonics
It may be writen in cylindrical coordinates $(rho, vartheta, z)$ in expansion (see Section 3.11 of Jackson book "Classical Electrodynamics") using Bessel function
According to wikipedia, expansions also exist in
- Bispherical coordinates
- Toroidal coordinates
- Prolate spheroidal coordinates
- Oblate spheroidal coordinates
Question: Are these expansions well known? What is references?
coordinate-systems laplacian greens-function electromagnetism
asked Dec 31 '18 at 13:17
Nigel1Nigel1
12010
$endgroup$
add a comment |
$begingroup$
Green's function for the three-variable Laplace equation in Cartesian coordinates is
$$frac{1}{|mathbf{r}-mathbf{r'}|} = frac{1}{sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}$$
It may be written in spherical coordinates $(r, theta, phi)$ in Laplace expansion (see here or Section 3.9 of Jackson book "Classical Electrodynamics") using spherical harmonics
It may be writen in cylindrical coordinates $(rho, vartheta, z)$ in expansion (see Section 3.11 of Jackson book "Classical Electrodynamics") using Bessel function
According to wikipedia, expansions also exist in
- Bispherical coordinates
- Toroidal coordinates
- Prolate spheroidal coordinates
- Oblate spheroidal coordinates
Question: Are these expansions well known? What is references?
coordinate-systems laplacian greens-function electromagnetism
asked Dec 31 '18 at 13:17
Nigel1Nigel1
12010
$endgroup$
add a comment |
$begingroup$
Green's function for the three-variable Laplace equation in Cartesian coordinates is
$$frac{1}{|mathbf{r}-mathbf{r'}|} = frac{1}{sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}$$
It may be written in spherical coordinates $(r, theta, phi)$ in Laplace expansion (see here or Section 3.9 of Jackson book "Classical Electrodynamics") using spherical harmonics
It may be writen in cylindrical coordinates $(rho, vartheta, z)$ in expansion (see Section 3.11 of Jackson book "Classical Electrodynamics") using Bessel function
According to wikipedia, expansions also exist in
- Bispherical coordinates
- Toroidal coordinates
- Prolate spheroidal coordinates
- Oblate spheroidal coordinates
Question: Are these expansions well known? What is references?
coordinate-systems laplacian greens-function electromagnetism
asked Dec 31 '18 at 13:17
Nigel1Nigel1
12010
$endgroup$
Green's function for the three-variable Laplace equation in Cartesian coordinates is
$$frac{1}{|mathbf{r}-mathbf{r'}|} = frac{1}{sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}$$
It may be written in spherical coordinates $(r, theta, phi)$ in Laplace expansion (see here or Section 3.9 of Jackson book "Classical Electrodynamics") using spherical harmonics
It may be writen in cylindrical coordinates $(rho, vartheta, z)$ in expansion (see Section 3.11 of Jackson book "Classical Electrodynamics") using Bessel function
According to wikipedia, expansions also exist in
- Bispherical coordinates
- Toroidal coordinates
- Prolate spheroidal coordinates
- Oblate spheroidal coordinates
Question: Are these expansions well known? What is references?
coordinate-systems laplacian greens-function electromagnetism
coordinate-systems laplacian greens-function electromagnetism
asked Dec 31 '18 at 13:17
Nigel1Nigel1
12010
asked Dec 31 '18 at 13:17
Nigel1Nigel1
12010
asked Dec 31 '18 at 13:17
Nigel1Nigel1
12010
asked Dec 31 '18 at 13:17
Nigel1Nigel1
12010
asked Dec 31 '18 at 13:17
Nigel1Nigel1
12010
12010
add a comment |
add a comment |
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