MHB Change of variables/ Transformations part 2

[Joe20]

I am not sure how I should set my u and v expressions into the u-v plane for this question.
How should I look at the expression to set u and v expressions?

 

[Opalg]

I am not sure how I should set my u and v expressions into the u-v plane for this question.
How should I look at the expression to set u and v expressions?

You could start by letting $x = 2u$ and $y = 3v$. You might then want to make a further change, from cartesian to polar coordinates.

 

[HOI]

You probably know that parametric equations for a circle with radius r, centered at (0, 0), x^2+ y^2= r^2, are x= r cos(t), y= r sin(t) because x^2+ y^2= r^2cos^2(t)+ r^2 sin^2(t)= r^2(cos^2(t)+ sin^2(t))= r^2.

It should not be too much of a "jump" to see that parametric equations for the ellipse, with axes of length a and b in the x and y direction, respectively, x^2/a^2+ y^2/b^2= 1, are x= a cos(t), y= b sin(t).