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(20 points) In the integral T? +v" "rvdndy I= [ 1 +(2-V)e make the change of variables = 2_v= U = %IVHint: Use the Jacobian and evaluate triple integral ...

Question

(20 points) In the integral T? +v" "rvdndy I= [ 1 +(2-V)e make the change of variables = 2_v= U = %IVHint: Use the Jacobian and evaluate triple integral J2-0J-8= dydrdz.

(20 points) In the integral T? +v" "rvdndy I= [ 1 +(2-V)e make the change of variables = 2_v= U = %IV Hint: Use the Jacobian and evaluate triple integral J2-0 J-8= dydrdz.



Answers

Compute the Jacobian (at the point, if indicated).
$$G(u, v)=\left(v \ln u, u^{2} v^{-1}\right), \quad(u, v)=(1,2)$$

All right. So we're supposed to calculate the Jacoby in for the given transformation. So let's go through. Conquer the partial derivatives. A partial X district to you is to be partial X prospective e is to you partial. Why respect to you is to you partial vice for TV is negative. Two V botched the negative there. So the determinant that is negative for the squared minus for you squared and we're done.

Appropriate. 13 Let G u F M V is equal to three u plus four Be a new minus two beat. So jack off G is equal to determinant off portion portion you for partial X portion you own system, which is three um, partial x partial v for this term, which is for and partial why partial you for this term, which is one on partial. Why partial B which is negative. Two eso determined here is three times negative two minus four times one, which is negative.

Problem. 18 Jack. Uh huh. G is equal to the determinant off partial X portion you, which is E power bi. Um, partial X portion V, which is you? E power bi abortion. Why fortune you? Which is he? Power you and partial. Why, unfortunately which is equal to zero. So they determine is this item. I'm not buying this coupon Interest is the Roman that this item, the mind these items is negative. You be for you cross the

I was Just find the country cobian for the given transformation X equals U plus view of square to wise equal to u minus. Fear was heard to So the Jacoby in is gonna be equal to Sorry. Okay. Unequal to partial of ex respect to you is one over the square to partial extractive e is one over the square to two partial y. With respect to you is one over squared to partial wise work to be is negative one over the square to That means that the determinant is going to be negative 1/2 minus a positive 1/2. And that is simply going to be negative one and we are done.


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