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YZ
Numerade Educator

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Problem 2 Medium Difficulty

a. Solve the system
$$u=x+2 y, \quad v=x-y$$
for $x$ and $y$ in terms of $u$ and $v .$ Then find the value of the Jacobian $\partial(x, y) / \partial(u, v)$
\begin{equation}
\begin{array}{l}{\text { b. Find the image under the transformation } u=x+2 y} \\ {v=x-y \text { of the triangular region in the } x y \text { -plane bounded }} \\ {\text { by the lines } y=0, y=x, \text { and } x+2 y=2 . \text { Sketch the trans- }} \\ {\text { formed region in the } u v \text { -plane. }}\end{array}
\end{equation}

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Final Answer given in solution

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Video Transcript

So first we're given that you request or explodes too. Why? And, uh because two x minus y. So we're gonna so excellent war for you will be, um this equals to this. This implies exit was who X equals to you. Plus two b over three and the workers to leam honest to you over three. So a simple equation. And now we're gonna find that rico been the geico been for. That's why I overdosed a UV. It's simple, really. So that's over w that sort over that way. The Why over dealt you and tell her why over that we the answer for that is 1/3 it's party. Ethel Carby. We're gonna transform the region from the O X Y plane to the X Y plane to the U O V plea. Let's see. Um so why cause to eggs employs X minus y zero. So that in plus the zero and also why he goes to this issue is zero. Also, we have voice to s, and this employs V zero. So we have you and the are zero. That's gonna be the U excess and be re excess. And for the last port, we had a line segment X equals who won and which employs X minus Y plus two x plus Y was three. And recall that the range for exist ones from 112 one That too. So what's gonna be the range for that of you envy? So here we have your class. We was three. So that's gonna be exactly from Ah 03230 on over you and the plane. So the region is a triangle with and then two degree angle and, uh, to size with long. Three. Okay, that's the region for over you and B.

YZ
University of Illinois at Urbana-Champaign
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