Solutions to $Delta u = u_xx + u_yy + u_zz = 0$ that only depend on r The 2019 Stack Overflow Developer Survey Results Are InProving that the bi-laplacian of a radial basis function is the dirac deltaLaplacian $Delta u$ in spherical coordinatesSolutions of the Laplace Equation in spherical coordinatesDoes particular+homogeneous capture all solutions for a linear pde?Laplacian in polar coordinates (idea)why is the laplacian of 1/r equal zero outside the origin?Laplace equation - boundary value problemApparent Contradictory Algebra in the Derivation of the Laplacian in Spherical CoordinatesSolve Laplace's equation in spherical coordinates, $nabla^2 u(r,theta,phi)=0$, in the general case.Deriving an Explicit Formula using Fundamental Solutions
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Solutions to $Delta u = u_xx + u_yy + u_zz = 0$ that only depend on r
The 2019 Stack Overflow Developer Survey Results Are InProving that the bi-laplacian of a radial basis function is the dirac deltaLaplacian $Delta u$ in spherical coordinatesSolutions of the Laplace Equation in spherical coordinatesDoes particular+homogeneous capture all solutions for a linear pde?Laplacian in polar coordinates (idea)why is the laplacian of 1/r equal zero outside the origin?Laplace equation - boundary value problemApparent Contradictory Algebra in the Derivation of the Laplacian in Spherical CoordinatesSolve Laplace's equation in spherical coordinates, $nabla^2 u(r,theta,phi)=0$, in the general case.Deriving an Explicit Formula using Fundamental Solutions
$begingroup$
Find all the solutions of $Delta u = u_xx + u_yy + u_zz = 0$ in
three dimensions that depend only on $r = x^2 + y^2 + z^2$, the
radial variable in polar coordinates. Use the following formula for ∆
in spherical polar coordinates $$u_rr + frac2ru_r +
frac1r^2u_theta theta + frac1r^2cot(theta)u_theta +
frac1r^2 (sintheta)^2u_phi phi $$
I'm not sure how to start this problem. I'm thinking that $u_rr + frac2ru_r = 0$ if $u$ is only to depend on r? Not sure how to continue
pde harmonic-functions laplacian
asked Apr 8 at 0:53
meff11meff11
566
$endgroup$
add a comment |
$begingroup$
Find all the solutions of $Delta u = u_xx + u_yy + u_zz = 0$ in
three dimensions that depend only on $r = x^2 + y^2 + z^2$, the
radial variable in polar coordinates. Use the following formula for ∆
in spherical polar coordinates $$u_rr + frac2ru_r +
frac1r^2u_theta theta + frac1r^2cot(theta)u_theta +
frac1r^2 (sintheta)^2u_phi phi $$
I'm not sure how to start this problem. I'm thinking that $u_rr + frac2ru_r = 0$ if $u$ is only to depend on r? Not sure how to continue
pde harmonic-functions laplacian
asked Apr 8 at 0:53
meff11meff11
566
$endgroup$
1
$begingroup$
Yeah, that's how you start. If $u$ depends only on $r$, then $u_theta = u_phi = 0$. This is an ODE that you can solve a variety of ways.
$endgroup$
– Neal
Apr 8 at 1:05
add a comment |
$begingroup$
Find all the solutions of $Delta u = u_xx + u_yy + u_zz = 0$ in
three dimensions that depend only on $r = x^2 + y^2 + z^2$, the
radial variable in polar coordinates. Use the following formula for ∆
in spherical polar coordinates $$u_rr + frac2ru_r +
frac1r^2u_theta theta + frac1r^2cot(theta)u_theta +
frac1r^2 (sintheta)^2u_phi phi $$
I'm not sure how to start this problem. I'm thinking that $u_rr + frac2ru_r = 0$ if $u$ is only to depend on r? Not sure how to continue
pde harmonic-functions laplacian
asked Apr 8 at 0:53
meff11meff11
566
$endgroup$
Find all the solutions of $Delta u = u_xx + u_yy + u_zz = 0$ in
three dimensions that depend only on $r = x^2 + y^2 + z^2$, the
radial variable in polar coordinates. Use the following formula for ∆
in spherical polar coordinates $$u_rr + frac2ru_r +
frac1r^2u_theta theta + frac1r^2cot(theta)u_theta +
frac1r^2 (sintheta)^2u_phi phi $$
I'm not sure how to start this problem. I'm thinking that $u_rr + frac2ru_r = 0$ if $u$ is only to depend on r? Not sure how to continue
pde harmonic-functions laplacian
pde harmonic-functions laplacian
asked Apr 8 at 0:53
meff11meff11
566
asked Apr 8 at 0:53
meff11meff11
566
asked Apr 8 at 0:53
meff11meff11
566
asked Apr 8 at 0:53
meff11meff11
566
asked Apr 8 at 0:53
meff11meff11
566
566
1
$begingroup$
Yeah, that's how you start. If $u$ depends only on $r$, then $u_theta = u_phi = 0$. This is an ODE that you can solve a variety of ways.
$endgroup$
– Neal
Apr 8 at 1:05
add a comment |
1
$begingroup$
Yeah, that's how you start. If $u$ depends only on $r$, then $u_theta = u_phi = 0$. This is an ODE that you can solve a variety of ways.
$endgroup$
– Neal
Apr 8 at 1:05
1
1
$begingroup$
Yeah, that's how you start. If $u$ depends only on $r$, then $u_theta = u_phi = 0$. This is an ODE that you can solve a variety of ways.
$endgroup$
– Neal
Apr 8 at 1:05
$begingroup$
Yeah, that's how you start. If $u$ depends only on $r$, then $u_theta = u_phi = 0$. This is an ODE that you can solve a variety of ways.
$endgroup$
– Neal
Apr 8 at 1:05
add a comment |
0
active
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$begingroup$
Yeah, that's how you start. If $u$ depends only on $r$, then $u_theta = u_phi = 0$. This is an ODE that you can solve a variety of ways.
$endgroup$
– Neal
Apr 8 at 1:05