Please help with PDE in polar coordinates.

For Polar coordinates, $\displaystyle r^2=x^2+y^2$ and $\displaystyle x=r\cos \theta$, $\displaystyle y=r\sin\theta$

$\displaystyle r^2=x^2+y^2\Rightarrow\; r\frac{\partial {r}}{\partial {x}}=x+y\frac{\partial {y}}{\partial {x}}$

$\displaystyle \Rightarrow\; \frac{\partial {r}}{\partial {x}}=\frac{x}{r}+\frac{y}{r}\frac{\partial {y}}{\partial {x}}$

$\displaystyle \frac{\partial {y}}{\partial {x}}=0$, then $\displaystyle \frac{\partial {r}}{\partial {x}}=\frac{x}{r}$

$\displaystyle \frac{\partial {r}}{\partial {x}}=\cos\theta$ as $\displaystyle \frac{\partial {y}}{\partial {x}}=0 $

You can now put in $\displaystyle 90^o\;>\;\theta\;>45^o$, you'll get $\displaystyle r>x$ and $\displaystyle \frac{\partial {r}}{\partial {x}}>1$

Let's just use an example where ##\theta =60^o##, so to every unit change of $\displaystyle x$, $\displaystyle r$will change for 2 unit. So $\displaystyle \frac{\partial {r}}{\partial {x}}=2$

The same reasoning, $\displaystyle r=2x$. So using this example, $\displaystyle \frac{\partial {r}}{\partial {x}}=\cos 60^o=0.5$ which does not agree with the example I gave.

I am missing something, please help.

Re: Please help with PDE in polar coordinates.

I am puzzled as to what your question is . Your title refers to a "PDE in polar coordinates" but there is no PDE, just relations between Cartesian Coordinates and Polar Coordinates. In any case, from $\displaystyle x= r cos(\theta)$, $\displaystyle r= \dfrac{x}{cos(\theta)}$ so $\displaystyle \dfrac{\partial r}{\partial x}= \dfrac{1}{cos(\theta)}$, not $\displaystyle cos(\theta)$. Similarly, $\displaystyle \dfrac{\partial r}{\partial x}= \dfrac{r}{x}= \dfrac{r}{rcos(\theta)}= \dfrac{1}{cos(\theta)}$.

Re: Please help with PDE in polar coordinates.

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Originally Posted by

**HallsofIvy** I am puzzled as to what your question is . Your title refers to a "PDE in polar coordinates" but there is no PDE, just relations between Cartesian Coordinates and Polar Coordinates. In any case, from $\displaystyle x= r cos(\theta)$, $\displaystyle r= \dfrac{x}{cos(\theta)}$ so $\displaystyle \dfrac{\partial r}{\partial x}= \dfrac{1}{cos(\theta)}$, not $\displaystyle cos(\theta)$. Similarly, $\displaystyle \dfrac{\partial r}{\partial x}= \dfrac{r}{x}= \dfrac{r}{rcos(\theta)}= \dfrac{1}{cos(\theta)}$.

Thanks for the reply, but according to the derivation shown in my first post, $\displaystyle \frac{\partial {r}}{\partial {x}}=\frac {x}{r}=\cos\theta$

This is verified by more than one book.

Re: Please help with PDE in polar coordinates.

Quote:

Originally Posted by

**Alan0354** For Polar coordinates, $\displaystyle r^2=x^2+y^2$ and $\displaystyle x=r\cos \theta$, $\displaystyle y=r\sin\theta$

$\displaystyle r^2=x^2+y^2\Rightarrow\; r\frac{\partial {r}}{\partial {x}}=x+y\frac{\partial {y}}{\partial {x}}$

$\displaystyle \Rightarrow\; \frac{\partial {r}}{\partial {x}}=\frac{x}{r}+\frac{y}{r}\frac{\partial {y}}{\partial {x}}$

$\displaystyle \frac{\partial {y}}{\partial {x}}=0$, then $\displaystyle \frac{\partial {r}}{\partial {x}}=\frac{x}{r}$

$\displaystyle \frac{\partial {r}}{\partial {x}}=\cos\theta$ as $\displaystyle \frac{\partial {y}}{\partial {x}}=0 $

You can now put in $\displaystyle 90^o\;>\;\theta\;>45^o$, you'll get $\displaystyle r>x$ and $\displaystyle \frac{\partial {r}}{\partial {x}}>1$

Let's just use an example where ##\theta =60^o##, so to every unit change of $\displaystyle x$, $\displaystyle r$will change for 2 unit. So $\displaystyle \frac{\partial {r}}{\partial {x}}=2$

The same reasoning, $\displaystyle r=2x$. So using this example, $\displaystyle \frac{\partial {r}}{\partial {x}}=\cos 60^o=0.5$ which does not agree with the example I gave.

I am missing something, please help.

$\displaystyle \frac{\partial {r}}{\partial {x}}=\cos\theta$ if $\displaystyle \frac{\partial {y}}{\partial {x}}=0$. If $\displaystyle r=2x=\sqrt{x^2+y^2}$, then $\displaystyle y = \pm x\sqrt{3}$, so $\displaystyle \frac{\partial {y}}{\partial {x}} = \pm \sqrt{3} \neq 0$.

Re: Please help with PDE in polar coordinates.

Quote:

Originally Posted by

**SlipEternal** $\displaystyle \frac{\partial {r}}{\partial {x}}=\cos\theta$ if $\displaystyle \frac{\partial {y}}{\partial {x}}=0$. If $\displaystyle r=2x=\sqrt{x^2+y^2}$, then $\displaystyle y = \pm x\sqrt{3}$, so $\displaystyle \frac{\partial {y}}{\partial {x}} = \pm \sqrt{3} \neq 0$.

Thanks for the reply. This is just so confusing.

$\displaystyle \frac{\partial y}{\partial x}=\frac{\partial y(r,\theta)}{\partial r}\frac{\partial r}{\partial x(r,\theta)}+\frac{\partial y(r,\theta)}{\partial \theta}\frac{\partial \theta}{\partial x(r,\theta)}=\cos\theta \frac{\partial y(r,\theta)}{\partial r}-\frac{\sin\theta}{r}\frac{\partial y(r,\theta)}{\partial \theta }= 0 $

So what is correct?

Re: Please help with PDE in polar coordinates.

Just a thought...are we to assume that you have a DEq that is being integrated over a region? And that you are trying to convert that region from rectangular coordinates to polar coordinates? In that case x and y are independent (except on the boundaries) and thus $\displaystyle \frac{\partial y}{\partial x} = 0$.

It might help to post the entire problem.

-Dan

Re: Please help with PDE in polar coordinates.

Quote:

Originally Posted by

**topsquark** Just a thought...are we to assume that you have a DEq that is being integrated over a region? And that you are trying to convert that region from rectangular coordinates to polar coordinates? In that case x and y are independent (except on the boundaries) and thus $\displaystyle \frac{\partial y}{\partial x} = 0$.

It might help to post the entire problem.

-Dan

Thanks for the reply. This is the complete question. I just find this that I cannot explain. It is just simple Polar coordinates and I listed the derivation according to more than one sources in the first post. Then I just use an example to show a different result that I cannot explain.

As you can see from the derivation, $\displaystyle \frac{\partial{r}}{\partial{x}}=\frac{x}{r}$ if $\displaystyle \frac{\partial{y}}{\partial{x}}=0$ as I showed my work. But that does not explain the example I gave when $\displaystyle 45^o<\theta<90^o$.

And the plot even thickens when SlipEternal showed that $\displaystyle \frac{\partial{y}}{\partial{x}}\neq 0$

Thanks

Re: Please help with PDE in polar coordinates.

Quote:

Originally Posted by

**Alan0354** Thanks for the reply. This is the complete question. I just find this that I cannot explain. It is just simple Polar coordinates and I listed the derivation according to more than one sources in the first post. Then I just use an example to show a different result that I cannot explain.

As you can see from the derivation, $\displaystyle \frac{\partial{r}}{\partial{x}}=\frac{x}{r}$ if $\displaystyle \frac{\partial{y}}{\partial{x}}=0$ as I showed my work. But that does not explain the example I gave when $\displaystyle 45^o<\theta<90^o$.

And the plot even thickens when SlipEternal showed that $\displaystyle \frac{\partial{y}}{\partial{x}}\neq 0$

Thanks

Well, that's why I'm curious. First about the DEq tag in your thread title. Why is that there if you don't have a DEq you are working with? And your comment that you found that $\displaystyle \frac{\partial y}{\partial x} = 0$ "verified by more than one book" leads me to believe you are converting a domain in xy to a domain in $\displaystyle r \theta$. At the very least you don't seem to be converting an equation (differential or otherwise) in this thread.

This whole question is just confusing!

-Dan

Re: Please help with PDE in polar coordinates.

Quote:

Originally Posted by

**topsquark** Well, that's why I'm curious. First about the DEq tag in your thread title. Why is that there if you don't have a DEq you are working with? And your comment that you found that $\displaystyle \frac{\partial y}{\partial x} = 0$ "verified by more than one book" leads me to believe you are converting a domain in xy to a domain in $\displaystyle r \theta$. At the very least you don't seem to be converting an equation (differential or otherwise) in this thread.

This whole question is just confusing!

-Dan

I might have use the wrong tittle, but my question is still valid. I triple checked my post before I posted it. Maybe, you can change it to a better tittle for me.

Thanks

Re: Please help with PDE in polar coordinates.

Now I am officially lost!!! I since posted this question in two different math forums, people there both asked and clarified, then no response!!!

I have one book and at least one article derived the equations like in my first post, but obviously the example I gave does not agree with the first post and I triple checked my example. I don't think I did anything wrong.......apparently I have not get any suggestion otherwise from three forums!!! I am pretty sure I am missing something as the book I used is a text book used in San Jose State and I studied through 7 chapters and yet to find a single mistake until the question here.

Anyone has anything to say?

From my example,

$\displaystyle \frac{\partial{r}}{\partial{x}}=\frac{r}{x}=\frac{ 1}{\cos\theta}$

And this answer makes a lot more sense.

Thanks

Re: Please help with PDE in polar coordinates.

Quote:

Originally Posted by

**Alan0354** Now I am officially lost!!! I since posted this question in two different math forums, people there both asked and clarified, then no response!!!

I have one book and at least one article derived the equations like in my first post, but obviously the example I gave does not agree with the first post and I triple checked my example. I don't think I did anything wrong.......apparently I have not get any suggestion otherwise from three forums!!! I am pretty sure I am missing something as the book I used is a text book used in San Jose State and I studied through 7 chapters and yet to find a single mistake until the question here.

Anyone has anything to say?

From my example,

$\displaystyle \frac{\partial{r}}{\partial{x}}=\frac{r}{x}=\frac{ 1}{\cos\theta}$

And this answer makes a lot more sense.

Thanks

The reason no one has a suggestion for you is because you are not asking any valid questions. If $\displaystyle \frac{\partial{y}}{\partial{x}}=0$, then $\displaystyle \frac{\partial{r}}{\partial{x}} = \cos \theta$. But, if $\displaystyle \frac{\partial{y}}{\partial{x}}\neq 0$, then $\displaystyle \frac{\partial{r}}{\partial{x}} \neq \cos \theta$.

Also, as HallsofIvy showed, $\displaystyle \frac{\partial{r}}{\partial{x}} = \dfrac{1}{\cos\theta}$, so setting those two equal, we get $\displaystyle \cos^2\theta = 1$, so $\displaystyle \frac{\partial{y}}{\partial{x}}=0$ only if $\displaystyle \theta = n\pi$ where $\displaystyle n$ is an integer. Hence, $\displaystyle \frac{\partial{y}}{\partial{x}}=0$ only if $\displaystyle \frac{\partial{r}}{\partial{x}} = \pm 1$.

Re: Please help with PDE in polar coordinates.

Quote:

Originally Posted by

**SlipEternal** The reason no one has a suggestion for you is because you are not asking any valid questions. If $\displaystyle \frac{\partial{y}}{\partial{x}}=0$, then $\displaystyle \frac{\partial{r}}{\partial{x}} = \cos \theta$. But, if $\displaystyle \frac{\partial{y}}{\partial{x}}\neq 0$, then $\displaystyle \frac{\partial{r}}{\partial{x}} \neq \cos \theta$.

Also, as HallsofIvy showed, $\displaystyle \frac{\partial{r}}{\partial{x}} = \dfrac{1}{\cos\theta}$, so setting those two equal, we get $\displaystyle \cos^2\theta = 1$, so $\displaystyle \frac{\partial{y}}{\partial{x}}=0$ only if $\displaystyle \theta = n\pi$ where $\displaystyle n$ is an integer. Hence, $\displaystyle \frac{\partial{y}}{\partial{x}}=0$ only if $\displaystyle \frac{\partial{r}}{\partial{x}} = \pm 1$.

But I posted in post #5 proof $\displaystyle \frac{\partial{y}}{\partial{x}}= 0$ for polar coordinates.

On top of that, you showed $\displaystyle \frac{\partial{y}}{\partial{x}}\neq 0$, which I agree with you. But the proof in post #5 say otherwise, so what am I missing?

Re: Please help with PDE in polar coordinates.

Quote:

Originally Posted by

**Alan0354** But I posted in post #5 proof $\displaystyle \frac{\partial{y}}{\partial{x}}= 0$ for polar coordinates.

On top of that, you showed $\displaystyle \frac{\partial{y}}{\partial{x}}\neq 0$, which I agree with you. But the proof in post #5 say otherwise, so what am I missing?

In post 5, you assumed $\displaystyle \frac{\partial{x(r,\theta)}}{\partial{r}} = \cos\theta$, which means you were already assuming $\displaystyle \frac{\partial{y}}{\partial{x}} = 0$. You never proved it.

Re: Please help with PDE in polar coordinates.

Quote:

Originally Posted by

**SlipEternal** In post 5, you assumed $\displaystyle \frac{\partial{x(r,\theta)}}{\partial{r}} = \cos\theta$, which means you were already assuming $\displaystyle \frac{\partial{y}}{\partial{x}} = 0$. You never proved it.

Oops, I wrote that upside down. I meant you assumed $\displaystyle \frac{\partial{r}}{\partial{x(r,\theta)}} = \cos\theta$

Re: Please help with PDE in polar coordinates.

Quote:

Originally Posted by

**SlipEternal** Oops, I wrote that upside down. I meant you assumed $\displaystyle \frac{\partial{r}}{\partial{x(r,\theta)}} = \cos\theta$

I think you find my problem. Let me work this out:

$\displaystyle r^2=x^2+y^2\;\Rightarrow\;\frac{\partial{r}}{\part ial{x}}=\frac {x}{r}+\frac{y}{r}\frac{\partial{y}}{\partial{x}}$

$\displaystyle x=r\cos\theta\;\Rightarrow \partial{x}=\partial{r}\cos\theta\;\Rightarrow\;\f rac{\partial{r}}{\partial{x}}=\frac{1}{\cos\theta} =\frac{r}{x}$

$\displaystyle \Rightarrow\;\frac{r}{x}=\frac {x}{r}+\frac{y}{r}\frac{\partial{y}}{\partial{x}}$

$\displaystyle \Rightarrow\;\frac{\partial{y}}{\partial{x}}=\left (\frac{r}{x}-\frac{x}{r}\right)\frac{r}{y}$

$\displaystyle y=r\sin\theta$ and $\displaystyle x=r\cos\theta$

$\displaystyle \Rightarrow\;\frac{\partial{y}}{\partial{x}}=\left (\frac{r}{x}-\frac{x}{r}\right)\frac{r}{y} = \frac{r^2}{r^2 \cos\theta \sin\theta}-\frac{\cos\theta}{\sin\theta}=\tan\theta$

I really appreciate your help, I know it must be my mistake, I don't believe for a moment the book is wrong. Just small little mistake cause me over a day!!! Now everything comes together.

Thanks