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1. The problem statement, all variables and given/known data

1. Consider the transformation x=u^{2}+v^{2}and y=u^{3}+v^{3}.

(a) Check that the Jacobian equals 0 on the line v=u and on the u-axis and on the v-axis.

(b) Show that the transform of the line v=u is the curve 2y^{2}=x^{3}, and that the transformation of the u-axis and v-axis form the curve y^{2}=x^{3}. Draw those curves.

(c) Find the regions in the u-v plane where the Jacobian of the transofmation is positive and the regions where the Jacobian is negative.

(d) Show that the transformation is 2:1 to the region in the x-y plane inside the curve 2y^{2}=x^{3}, and is 4:1 to the region in the x-y plane between the curves 2y^{2}=x^{3}and y^{2}=x^{3}, using the sign information for the Jacobian in (c) and the fact about the direction of motion just cited.

2. Relevant equations

Suppose that the transformation transforms some circle about a point P in the u-v plane into a loop about the image of P in the x-y plane, and that the region inside the circle is transformed into the region inside the loop. We will show in class that, at a point on the circle where the Jacobian of the transformation is positive, when you move counterclockwise about the circle, then you are moving counterclockwise about the image loop.

3. The attempt at a solution

(a), (b), and (c) were a cinch. Just so you know, my graph on (c) is the line u=v in the u-v plane. I indicated that the Jacobian was positive in quadrant 4 and negative in quadrant 2; and that in is positive in top slice of quadrants 1 and 3 and negative in the bottom slice of those same quadrants (I hope that makes sense!).

Anyways, (d) is giving me trouble. I can graph the regions in the x-y plane pretty easily, but not the corresponding regions in the u-v plane. For example, I want to show that the region inside 2y^{2}=x^{3}goes to two different regions in the in the u-v plane. It's getting really complicated and annoying.

2(u^{3}+v^{3})^{2}=(u^{2}+v^{2})^{3}

2(u^{6}+2u^{3}v^{3}+v^{6})=(u^{4}+2u^{2}v^{2}+v^{4})(u^{2}+v^{2})

2u^{6}+4u^{3}v^{3}+2v^{6}=u^{6}+2u^{4}v^{2}+u^{2}v^{4}+u^{4}v^{2}+2u^{2}v^{4}+v^{6}

u^{6}+4u^{3}v^{3}+v^{6}-3u^{4}v^{2}-3u^{2}v^{4}=0

.... and I don't even know if I did that right. Or if I'm doing this problem the right way.

Help, please.

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# Homework Help: Homework problem: Consider the transformation

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