Calculus 2nd Edition Rogawski

938 C H A P T E R 16 MULTIPLE INTEGRATION
8. Show that G maps the line v = mu to the line of slope
(5 + 3m)/(2 + m) through the origin in the xy-plane.
9. Show that the inverse of G is
G −1 (x, y) = (3x − y,−5x + 2y)
Hint: Show that G(G −1 (x, y)) = (x, y) and G −1 (G(u, v)) = (u, v).
10. Use the inverse in Exercise 9 to find:
(a) A point in the uv-plane mapping to (2, 1)
(b) A segment in the uv-plane mapping to the segment joining (−2, 1)
and (3, 4)
11. Calculate Jac(G) =
12. Calculate Jac(G −1 ) =
∂(x, y)
∂(u, v) .
∂(u, v)
∂(x, y) .
In Exercises 13–18, compute the Jacobian (at the point, if indicated).
13. G(u, v) = (3u + 4v, u − 2v)
14. G(r, s) = (rs, r + s)
15. G(r, t) = (r sin t,r − cos t), (r, t) = (1, π)
16. G(u, v) = (v ln u, u 2 v −1 ), (u, v) = (1, 2)
17. G(r, θ) = (r cos θ,rsin θ), (r, θ) = ( 4, π )
6
18. G(u, v) = (ue v ,e u )
19. Find a linear mapping G that maps [0, 1] × [0, 1] to the parallelogram
in the xy-plane spanned by the vectors ⟨2, 3⟩ and ⟨4, 1⟩.
20. Find a linear mapping G that maps [0, 1] × [0, 1] to the parallelogram
in the xy-plane spanned by the vectors ⟨−2, 5⟩ and ⟨1, 7⟩.
21. Let D be the parallelogram in Figure 13. Apply the Change of
Variables ∫∫ Formula to the map G(u, v) = (5u + 3v, u + 4v) to evaluate
xy dx dy as an integral over D 0 =[0, 1] × [0, 1].
D
(d) Observe that by the formula for the area of a triangle, the region D
in Figure 14 has area 1 2 (b2 − a 2 ). Compute this area again, using the
Change of Variables Formula applied to G.
∫∫
(e) Calculate xy dx dy.
D
b
a
y
D
a
FIGURE 14
23. Let G(u, v) = (3u + v, u − 2v). Use the Jacobian to determine
the area of G(R) for:
(a) R =[0, 3] × [0, 5] (b) R =[2, 5] × [1, 7]
24. Find a linear map T that maps [0, 1] × [0, 1] to the parallelogram P
in the xy-plane with vertices (0, 0), (2, 2), (1, 4), (3, 6). Then calculate
the double integral of e 2x−y over P via change of variables.
25. With G as in Example 3, use the Change of Variables Formula to
compute the area of the image of [1, 4] × [1, 4].
In Exercises 26–28, let R 0 =[0, 1] × [0, 1] be the unit square. The
translate of a map G 0 (u, v) = (φ(u, v), ψ(u, v)) is a map
G(u, v) = (a + φ(u, v), b + ψ(u, v))
where a,b are constants. Observe that the map G 0 in Figure 15 maps
R 0 to the parallelogram P 0 and that the translate
maps R 0 to P 1 .
G 1 (u, v) = (2 + 4u + 2v, 1 + u + 3v)
b
x
y
y
(3, 4)
D
FIGURE 13
(5, 1)
x
1
1
0
0
1
1
G 0 (u, ) = (4u + 2 , u + 3 )
u
G 1 (u, ) = (2 + 4u + 2 , 1 + u + 3 )
u
(2, 3)
0
y
(4, 4)
(6, 4)
(4, 1)
x
(8, 5)
1
(6, 2)
x
(2, 1)
22. Let G(u, v) = (u − uv, uv).
(a) Show that the image of the horizontal line v = c is y =
c ̸= 1, and is the y-axis if c = 1.
(b) Determine the images of vertical lines in the uv-plane.
(c) Compute the Jacobian of G.
c
1 − c x if
y
(4, 5)
(2, 2)
2
(6, 3)
x
(1, 4)
(−1, 1)
FIGURE 15
y
3
(3, 2)
x