SECTION 2.7 EXERCISES Review Questions 23-26. Solve and compute Jacobians Solve the following relations for 1. Suppose S is the unit square in the...
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Could use some help in solving problem 30 in preparation for my midterm

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SECTION 2.7 EXERCISES
Review Questions
23-26. Solve and compute Jacobians Solve the following relations for
1. Suppose S is the unit square in the first quadrant of the uv-plane.
x and y, and compute the Jacobian J(u, v).
Describe the image of the transformation T: x = 2u, y = 2v.
23. u = x + y, v = 2x - y
24. u = xy, v = x
2.
Explain how to compute the Jacobian of the transformation
T: x = 8(u, v), y = h(u, v).
25. u = 2x - 3y, v =y-x
26. u = x + 4y, v = 3x + 2y
3.
Using the transformation T: x = u + v, y = u - v, the image of
27-30. Double integrals-transformation given To evaluate the fol-
the unit square S = { (u, v): 0 S u = 1, 0 = v = 1} is a region
lowing integrals, carry out these steps.
in the xy-plane. Explain how to change variables in the integral
a. Sketch the original region of integration R in the xy-plane and the
JRf(x, y) dA to find a new integral over S.
new region S in the uv-plane using the given change of variables.
b. Find the limits of integration for the new integral with respect to u
4.
Suppose S is the unit cube in the first octant of uvw-space with
and v.
one vertex at the origin. What is the image of the transformation
c. Compute the Jacobian.
T: x = u/2, y = v/2, z = w/2?
d. Change variables and evaluate the new integral.
Basic Skills
5-12. Transforming a square Let S = { (u, v): 0 S us 1,
27 .
xy dA, where R is the square with vertices (0, 0), (1, 1),
O S v s 1} be a unit square in the uv-plane. Find the image of S in
(2, 0), and (1, -1); use x = u + v, y = u - v.
the xy-plane under the following transformations.
5. T: x = 2u, y = v/2
28 .
// x 3y dA , where R = { ( x , y ) : 0 5 x 5 2, x sysx+ 4);
6. T: x = -u,y = -v
use x = 2u, y = 4v + 2u.
7. T: x = (u + v)/2,y = (u-v)/2
8. T: x = 2u + v,y = 2u
29 .
x 2 Vx + 2y dA, where
9. 1:x = U2 - v2, y = 2uv
R
R = { ( x, y ) : 0 = x = 2, - x/ 2 = y = 1 - x}; use
10. T: x = 2uv, y = u2 - v2
x = 2u, y = v - u.
11. T: x = u Cos TV, y = u sin TV
30 .
xy da, where R is bounded by the ellipse 9x2 + 4y? = 36;
12. T: x = v sin Tru, y = V COS Tru
use x = 2u, y = 3v.
13-16. Images of regions Find the image R in the xy-plane of the re-
gion S using the given transformation T. Sketch both R and S.
31-36. Double integrals-your choice of transformation Evaluate
the following integrals using a change of variables. Sketch the original
13. S = { ( u, v) : v = 1 - u, u 2 0, v 2 0 }; T: x = u,y = v2
and new regions of integration, R and S.
14. S = { (u, v): u2 + v2 = 1}; 1:x= 2u,y=4v
15. S = { ( u, v) : 1 = u < 3, 2 = v = 4}; 1: x = u/v,y = v
31. J J Vx - vaxdy
16. S = { ( u, v ) : 2 = u < 3, 3 = v = 6}; 1: x = u,y = v/u
32 .
([ Vy2 - x2 dA, where R is the diamond bounded by
onily) .op
17-22. Computing Jacobians Compute the Jacobian J(u, v) for the
following transformations.
y - x = 0, y - x = 2, y+ x= 0, and y + x =2
17. T: x = 3u, y = -3v
33.
y - * dA, where R is the parallelogram bounded
18. T: x = 4v, y = -2u
// ( + 2x + 1)
R
19. T: x = 2uv, y = u2 - v2
by y - x = 1, y - x = 2, y + 2x = 0, and y + 2x = 4
20. T: x = u COS TTV, y = u sin TV
34.
ey dA, where R is the region bounded by the hyperbolas
21. T: x = (u+ v)/V2,y = (u-v)/V2
xy = 1 and xy = 4, and the lines y/ x = 1 and y/ x = 3
22. T: x = u/v,y = v

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