Change of variables and Jacobians

[jaredogden]

Homework Statement

I am confused on how exactly Jacobians variables and such. We had a problem on a test in my class that was:

a double integral (i don't know how to use the notation on here) over the region R of (x-2y)e^(x+y)dxdy Where R is the parallelogram with vertices (0,0) (2,1) (2,-2) (4,-1)

We were supposed to chose a change of variables, s and t, based on the integrand and region R. On the key after we got the test back she said s=x-2y and t=x+y

I am confused how exactly she knew that just by looking at the integrand that those were the equations to be used.. I can see graphically how they worked but I am lost as to how it was so easy to tell just from the integrand.

Also what is the correct method to properly convert the region to the new coordinate system based off of that? I can see mathematically how she got the answer for this part on the key online but I don't understand how she got the new graph based off s and t. Any help would be appreciated thanks!

P.S. I hope this was the proper area to post this. It's not exactly a homework question and it's an old test question.

Homework Equations

Jacobian f_x,y/f_s,t

The Attempt at a Solution

s=x-2y
t=x+y
Jacobian is (1/3)

 

[Char. Limit]

Well they are certainly the ones I would pick. Notice how they simplify the integral to an easy one, s e^t? She basically, probably, picked two that would simplify the integral.

 

[jaredogden]

Right I see that. But are there other equations that can be used? I just didn't understand how she knew what one was s and t and why x-2y couldn't be t and x+y couldn't be s. Is it simply just because they wouldn't work graphically? Maybe I just need to review more about Jacobian's and change of variables I guess..

 

[vela]

You don't look at the integrand to find the new variables; you look at the region R. In terms of the s and t coordinates, R is a rectangle, which makes writing the limits of integration easy. You then transform the integrand and hope it's straightforward to integrate.

 

[jaredogden]

Well on her key she has (x-2y) underlined and is says this suggests s=(x-2y) and (x+y) underlined and it says this suggests t=(x+y) so I was confused as to how that suggests that. So then how do you find the limits and everything from the region exactly?

 

[vela]

You can do it both ways. When I read "parallelogram," my first instinct was to change it into a rectangle, and that transformation would do that. If you focus instead on the integrand, as Char.Limit says, that choice makes it particularly simple. You could always try a different transformation, but why would you? The whole idea is to make the problem easier to do, so you start with the obvious choice and hope it pans out.

As far as the choice of which expression is s and which is t, it doesn't matter.

I think you're going to need to articulate exactly what is confusing you.

 

[Char. Limit]

You can do it both ways. When I read "parallelogram," my first instinct was to change it into a rectangle, and that transformation would do that. If you focus instead on the integrand, as Char.Limit says, that choice makes it particularly simple. You could always try a different transformation, but why would you? The whole idea is to make the problem easier to do, so you start with the obvious choice and hope it pans out.

As far as the choice of which expression is s and which is t, it doesn't matter.

I think you're going to need to articulate exactly what is confusing you.

To be honest, I don't even look at the region until I've made sure there isn't some transformation that would simplify the integrand enormously. But then, I do a bit of guess-and-check.

 

[jaredogden]

okay I'm going to try to explain what exactly is confusing me. I don't understand how vela said his first instinct was to change the parallelogram into a rectangle. I understand that Jacobians and change of variables are supposed to change regions of difficult integration into something simpler like simple rectangles and such, but I just didnt know how you go about figuring out the best equations for s and t to make the transformation. Like what if there was no e^(x+y) in the integrand then what would t equal?

Also does it matter in the original problem if s=x-2y and t=x+y? So basically if you can help me understand the steps to properly transform a region based off either the integrand and or the given points for the parallelogram. Does it have something to do with the equations of the lines that make up each side of the parallelogram?

I don't know if that helped at all, I think I'm confusing myself just thinking about it haha. Thanks for your help so far

 

[Char. Limit]

okay I'm going to try to explain what exactly is confusing me. I don't understand how vela said his first instinct was to change the parallelogram into a rectangle. I understand that Jacobians and change of variables are supposed to change regions of difficult integration into something simpler like simple rectangles and such, but I just didnt know how you go about figuring out the best equations for s and t to make the transformation. Like what if there was no e^(x+y) in the integrand then what would t equal?

Also does it matter in the original problem if s=x-2y and t=x+y? So basically if you can help me understand the steps to properly transform a region based off either the integrand and or the given points for the parallelogram. Does it have something to do with the equations of the lines that make up each side of the parallelogram?

I don't know if that helped at all, I think I'm confusing myself just thinking about it haha. Thanks for your help so far

You can pick either s or t for your variables. It'll just use different rectangles of the same area.

 

[vela]

Does it have something to do with the equations of the lines that make up each side of the parallelogram?

Yes. Try drawing the parallelogram on the xy plane and then plotting the lines s=0 and t=0 on the same diagram.

Keep in mind it's not really an either-or thing as far as whether you should look at the integrand or the region over which you're integrating. You should consider them both and try stuff that looks like it might work. You'll eventually develop some intuition on what to try based on doing a bunch of problems and seeing what works and what doesn't.