Hi, revising this subject i'm having problems with questions of this nature
Can anyone show me how to go about this?
I will assume the equation is, for .
Define by where and .
Now we will define a new function (of two variable) in the following way: where are the unique points such that . In other words, we define to be evaluated at a point to be evaluated at where is the (unique point in ) that is mapped into under the transformation that you gave.
It should be clear that i.e. (function composition) for each point .
Now we are going to use the chain rule. That is where I am going to be a little informal with notation because it gets cleaner to write this way and it perhaps be easier for you to follow.
Let us analyze what the first equation means (the second one is similar). It is saying that (by chain rule) we get the expression on the right hand side. When we write we mean the partial derivative with respect to the first coordinate evaluated at (that is at ) multiplied by - the partial derivative of the function with respect to the first coordinate. Similarly, means the partial derivative with respect to the second coordinate evaluated at (that is ).
Therefore, we get upon writing everything out,
Can you finish now comrade?
Hey pkr, can you post a plot for this? I think it would be cool. Just make up any old (reasonable) IBVP on it.
Can you do the second partials? What's it saying? What's it doing?
Why are so much phenomena in our world describable by second order PDEs?
Sumptin's up I tell you what.
Here's one of them:
Pretty sure that's right. Double check it.
I'll assume the mixed partials will drop out when everything is simplified, or else everything else will drop out and you're left with just one mixed partial. I may try to solve it (with just any old IBVP) but I'm slow so don't wait on me.