Question

6. If x = eucosv,y = eusinv then
= eusinv then is
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Answer 1

Answer:

You're almost there. First, a miscalculation: your ∂v∂x has the wrong sign. In particular: we indeed have

u=12log(x2+y2),v=tan−1(y/x).

It follows that

∂u∂x=xx2+y2,∂u∂y=yx2+y2,

which you calculated correctly. However,

∂v∂x=−yx2⋅11+(y/x)2=−yx2+y2,∂v∂y=1x⋅11+(y/x)2=xx2+y2.

Now, in order to see that JJ′=1, write both Jacobians as a function of the same set of variables. That is, either as a function of x and y or as a function of u and v.

If we write J′ as a function of u and v, then we find

J′=x2+y2(x2+y2)2=1x2+y2=1[eucosv]2+[eusinv]2=1e2u.

In greater detail: we have v(x,y)=tan−1(y/x). We find the derivative of this using the chain rule. Because ddttan−1(t)=11+t2, we have

∂v∂x=11+(y/x)2⋅∂∂x[(y/x)].

To calculate the partial derivative of y/x with respect to x, we use the power rule. In particular:

f(x,y)=y/x=y⋅x−1⟹∂f∂x=−yx−2=−y⋅1x2=−yx2.