Question

1/3 x + y 7. Consider dA where R is the region bounded by the triangle with vertices (0,0), (2,0), V= x+y X-y and (0,-2). The change of variables u=- defines a transformation T(x,y)=(u,v) from the xy-plane 2 to the uv-plane. (a) (10 pts) Write S (in terms of u and v) using set- builder notation, where T:R→S. Use T to help you sketch S in the uv-plane by evaluating T at the vertices. - 1 a(u,v) (b) (4 pts) Evaluate the determinant 2(x,y) 0 ах (c) (16 pts) The Jacobian a(x,y) a(u, v) = 1 VƏ(u,v) use this fact to evaluate 2(x,y)' 1/3 dA.

Answer 1

In this question we have to first find the set S it is nothing but we have to write in the stet notation then put the points one by one and hence again we have a triangle with different coordinates.

Now find all the partial derivatives as told and then find the determinant.

Then just take inverse and look at the region find the limits and integrate.

In he last result there is (-1)^(1/3) so it has he complex value if you want o see he real value of this then result is 0 but if complex then there is some value as $(-1)^{1/3}=e^{i\pi/3}$
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