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 , so , not . Similarly, .
For Polar coordinates, and ,
, then
as
You can now put in , you'll get and
Let's just use an example where ##\theta =60^o##, so to every unit change of , will change for 2 unit. So
The same reasoning, . So using this example, which does not agree with the example I gave.
I am missing something, please help.
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 , so , not . Similarly, .
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 .
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, if as I showed my work. But that does not explain the example I gave when .
And the plot even thickens when SlipEternal showed that
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 "verified by more than one book" leads me to believe you are converting a domain in xy to a domain in . 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
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,
And this answer makes a lot more sense.
Thanks