Hello, I've allways wondered how to get to polar coordinates from cartisan coordinates. I took a course in fluid mechanics but we never learned how to get the continuity equation from cartisan to polar. I know you can use physics to derive the polar equation, but I want to do it just by using mathematics and the cartisan equation.(adsbygoogle = window.adsbygoogle || []).push({});

In cartisan the equation is

[itex]\frac{\partial V_{x}}{\partial x}+\frac{\partial V_{y}}{\partial y}=0[/itex]

by using:

[itex]x=r*cos(\theta)\\

y = r*sin(\theta)[/itex]

I get:

[itex]

\begin{pmatrix}

\frac{dx}{dt} \\

\frac{dy}{dt}\end{pmatrix}

=

\begin{pmatrix}

V_{x} \\

V_{y}\end{pmatrix}

=

\begin{pmatrix}

cos(\theta) & -r*sin(\theta) \\

sin(\theta) & r*cos(\theta)\end{pmatrix}

*

\begin{pmatrix}

V_{r} \\

V_{\theta}\end{pmatrix}

[/itex], I have defined: [itex]

\begin{pmatrix}

\frac{dr}{dt} \\

\frac{d\theta}{dt}\end{pmatrix}

=

\begin{pmatrix}

V_{r} \\

V_{\theta}\end{pmatrix}

[/itex]

This gives:

[itex]V_{x}=cos(\theta)*V_{r}-r*sin(\theta)*V_{\theta}[/itex] and

[itex]V_{y}=sin(\theta)*V_{r}+r*cos(\theta)*V_{\theta}[/itex]

Now my problem arises, I do not see how I am supposed to calculate:

[itex]\frac{\partial V_{x}}{\partial x}+\frac{\partial V_{y}}{\partial y}=

\frac{\partial (cos(\theta)*V_{r}-r*sin(\theta)*V_{\theta})}{\partial x} +

\frac{ \partial (sin(\theta)*V_{r}+r*cos(\theta)*V_{\theta}) }{\partial y}

[/itex]

Can you guys help me how to end this? It is supposed to be at the end:

[itex]

\frac{1}{r}\frac{\partial}{\partial r}(r*V_{r})

+\frac{1}{r}\frac{\partial}{\partial \theta}(V_{\theta})=0

[/itex]

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# Continuity equation, cartisan to polar

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