Image after Transformation

**Rhode963** · March 30th 2010, 06:07 AM

Alright, so I've got to "Find the image of the set $S$ after the given transformation."

$S$ is the square bounded by the lines $u=0, u=1, v=0, v=1; x=v, y=u(1+v^2)$

How do I do this? I've found the Jacobian to be $-u$ but I don't even know how to use that for this problem. Can someone please start me in the correct direction.

**Rhode963** · March 30th 2010, 07:32 AM

I think I've figured it out. Basically I took all the existing boundry curves/lines, and converted them. The Jacobian wasn't used. But I ended up getting a figure that looks like the area underneath $y=x^2+1$ from 0 to 1. Does that sound correct?

**HallsofIvy** · March 30th 2010, 01:31 PM

Originally Posted by Rhode963

Alright, so I've got to "Find the image of the set $S$ after the given transformation."

$S$ is the square bounded by the lines $u=0, u=1, v=0, v=1; x=v, y=u(1+v^2)$

How do I do this? I've found the Jacobian to be $-u$ but I don't even know how to use that for this problem. Can someone please start me in the correct direction.

Since x= v, v= 0 and v= 1 become x= 0 and x= 1. The other sides are harder. Since $y= u(1+ v^2)$ and x= v, we can rewrite that as $y= u(1+ x^2)$ so that $u= \frac{y}{1+ x^2}$ . Now u= 0 becomes $\frac{y}{1+ x^2}= 0$ or $y= 0$ . Finally, $u= 1$ becomes $\frac{y}{1+ x^2}= 1$ or $y= 1+ x^2$ , a parabola. That's exactly what you have!

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