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Phase difference between variables in a PDE

Problem with Laplacian while treating polar coordinates as special case of spherical coordinates.Chain Rule in Polar coordinatesDot product of gradient and tangent vectorDifference of Entropy of two-dimensional Gaussians2nd Order PDE with Mixed DerivativesProof of vector identity from vorticity transport equationPDE Cartesian/Polar forms and Numerical Solver issuesLaplacian operator and divergence at the origin$nablaint_S_1(0) f(|x|w)dsigma(w) = int_S_1(0) nabla f(|x|w)dsigma(w)$Grad in polar coordinates

1

$begingroup$

For a given vector $boldsymbolu = (u,v,w)$, in a polar coordinate system $(x,r,theta)$, I have read that the divergence free equation:
begingather
nabla cdot boldsymbol u equiv
fracpartial upartial x + frac1rfracpartialpartial r(vr) + frac1rfracpartial wpartial theta = 0
endgather
shows that the phase difference between the components $u$ and $v$ is $pi/2$. How can I prove this?

I have tried representing each of the variables in polar form, and tried obtaining the phase difference between them but with no success.

complex-analysis pde polar-coordinates divergence

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asked 15 hours ago

XoegXoeg

62

New contributor

Xoeg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$

add a comment | 

1

$begingroup$

For a given vector $boldsymbolu = (u,v,w)$, in a polar coordinate system $(x,r,theta)$, I have read that the divergence free equation:
begingather
nabla cdot boldsymbol u equiv
fracpartial upartial x + frac1rfracpartialpartial r(vr) + frac1rfracpartial wpartial theta = 0
endgather
shows that the phase difference between the components $u$ and $v$ is $pi/2$. How can I prove this?

I have tried representing each of the variables in polar form, and tried obtaining the phase difference between them but with no success.

complex-analysis pde polar-coordinates divergence

share|cite|improve this question

asked 15 hours ago

XoegXoeg

62

New contributor

Xoeg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

For a given vector $boldsymbolu = (u,v,w)$, in a polar coordinate system $(x,r,theta)$, I have read that the divergence free equation:
begingather
nabla cdot boldsymbol u equiv
fracpartial upartial x + frac1rfracpartialpartial r(vr) + frac1rfracpartial wpartial theta = 0
endgather
shows that the phase difference between the components $u$ and $v$ is $pi/2$. How can I prove this?

I have tried representing each of the variables in polar form, and tried obtaining the phase difference between them but with no success.

complex-analysis pde polar-coordinates divergence

share|cite|improve this question

asked 15 hours ago

XoegXoeg

62

New contributor

Xoeg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

asked 15 hours ago

XoegXoeg

62

New contributor

Xoeg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

asked 15 hours ago

XoegXoeg

62

New contributor

Xoeg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 15 hours ago

XoegXoeg

62

New contributor

Xoeg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 15 hours ago

XoegXoeg

62