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Thread: Image after Transformation

  1. #1
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    Image after Transformation

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

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

    How do I do this? I've found the Jacobian to be $\displaystyle -u$ but I don't even know how to use that for this problem. Can someone please start me in the correct direction.
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  2. #2
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    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 $\displaystyle y=x^2+1$ from 0 to 1. Does that sound correct?
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  3. #3
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    Quote Originally Posted by Rhode963 View Post
    Alright, so I've got to "Find the image of the set $\displaystyle S$ after the given transformation."

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

    How do I do this? I've found the Jacobian to be $\displaystyle -u$ 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 $\displaystyle y= u(1+ v^2)$ and x= v, we can rewrite that as $\displaystyle y= u(1+ x^2)$ so that $\displaystyle u= \frac{y}{1+ x^2}$. Now u= 0 becomes $\displaystyle \frac{y}{1+ x^2}= 0$ or $\displaystyle y= 0$. Finally, $\displaystyle u= 1$ becomes $\displaystyle \frac{y}{1+ x^2}= 1$ or $\displaystyle y= 1+ x^2$, a parabola. That's exactly what you have!
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