14.7E: Exercises for Section 14.7
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In exercises 1 - 6, the function T:S→R, T(u,v)=(x,y) on the region S={(u,v)|0≤u≤1, 0≤v≤1} bounded by the unit square is given, where R∈R2 is the image of S under T.
a. Justify that the function T is a C1 transformation.
b. Find the images of the vertices of the unit square S through the function T.
c. Determine the image R of the unit square S and graph it.
1. x=2u, y=3v
2. x=u2, y=v3
- Answer
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a. T(u,v)=(g(u,v), h(u,v), x=g(u,v)=u2 and y=h(u,v)=v3. The functions g and h are continuous and differentiable, and the partial derivatives gu(u,v)=12, gv(u,v)=0, hu(u,v)=0 and hv(u,v)=13 are continuous on S;
b. T(0,0)=(0,0), T(1,0)=(12,0), T(0,1)=(0,13), and T(1,1)=(12,13);
c. R is the rectangle of vertices (0,0), (0,13), (12,13), and (0,13) in the xy-plane; the following figure.
3. x=u−v, y=u+v
4. x=2u−v, y=u+2v
- Answer
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a. T(u,v)=(g(u,v), h(u,v), x=g(u,v)=2u−v and y=h(u,v)=u+2v. The functions g and h are continuous and differentiable, and the partial derivatives gu(u,v)=2, gv(u,v)=−1, hu(u,v)=1 and hv(u,v)=2 are continuous on S;
b. T(0,0)=(0,0), T(1,0)=(2,1), T(0,1)=(−1,2), and T(1,1)=(1,3);
c. R is the parallelogram of vertices (0,0), (2,1) (1,3), and (−1,2) in the xy-plane; the following figure.
5. x=u2, y=v2
6. x=u3, y=v3
- Answer
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a. T(u,v)=(g(u,v), h(u,v), x=g(u,v)=u3 and y=h(u,v)=v3. The functions g and h are continuous and differentiable, and the partial derivatives gu(u,v)=3u2, gv(u,v)=0, hu(u,v)=0 and hv(u,v)=3v2 are continuous on S;
b. T(0,0)=(0,0), T(1,0)=(1,0), T(0,1)=(0,1), and T(1,1)=(1,1);
c. R is the unit square in the xy-plane see the figure in the answer to the previous exercise.
In exercises 7 - 12, determine whether the transformations T:S→R are one-to-one or not.
7. x=u2, y=v2, where S is the rectangle of vertices (−1,0), (1,0), (1,1), and (−1,1).
8. x=u4, y=u2+v, where S is the triangle of vertices (−2,0), (2,0), and (0,2).
- Answer
- T is not one-to-one: two points of S have the same image. Indeed, T(−2,0)=T(2,0)=(16,4).
9. x=2u, y=3v, where S is the square of vertices (−1,1), (−1,−1), (1,−1), and (1,1).
10. T(u,v)=(2u−v,u), where S is the triangle with vertices (−1,1),(−1,−1), and (1,−1).
- Answer
- T is one-to-one: We argue by contradiction. T(u1,v1)=T(u2,v2) implies 2u1−v1=2u2−v2 and u1=u2. Thus, u1=u+2 and v1=v2.
11. x=u+v+w, y=u+v, z=w, where S=R=R3.
12. x=u2+v+w, y=u2+v, z=w, where S=R=R3.
- Answer
- T is not one-to-one: T(1,v,w)=(−1,v,w)
In exercises 13 - 18, the transformations T:R→S are one-to-one. Find their related inverse transformations T−1:R→S.
13. x=4u, y=5v, where S=R=R2.
14. x=u+2v, y=−u+v, where S=R=R2.
- Answer
- u=x−2y3, v=x+y3
15. x=e2u+v, y=eu−v, where S=R2 and R={(x,y)|x>0, y>0}
16. x=lnu, y=ln(uv), where S={(u,v)|u>0, v>0} and R=R2.
- Answer
- u=ex, v=e−x+y
17. x=u+v+w, y=3v, z=2w, where S=R=R3.
18. x=u+v, y=v+w, z=u+w, where S=R=R3.
- Answer
- u=x−y+z2, v=x+y−z2, w=−x+y+z2
In exercises 19 - 22, the transformation T:S→R, T(u,v)=(x,y) and the region R⊂R2 are given. Find the region S⊂R2.
19. x=au, y=bv, R={(x,y)|x2+y2≤a2b2} where a,b>0
20. x=au, y=bc, R={(x,y)|x2a2+y2b2≤1}, where a,b>0
- Answer
- S={(u,v)|u2+v2≤1}
21. x=ua, y=vb, z=wc, R={(x,y)|x2+y2+z2≤1}, where a,b,c>0
22. x=au, y=bv, z=cw, R={(x,y)|x2a2−y2b2−z2c2≤1, z>0}, where a,b,c>0
- Answer
- R={(u,v,w)|u2−v2−w2≤1, w>0}
In exercises 23 - 32, find the Jacobian J of the transformation.
23. x=u+2v, y=−u+v
24. x=u32, y=vu2
- Answer
- 32
25. x=e2u−v, y=eu+v
26. x=uev, y=e−v
- Answer
- −1
27. x=u cos(ev), y=u sin(ev)
28. x=v sin(u2), y=v cos(u2)
- Answer
- 2uv
29. x=u coshv, y=u sinhv, z=w
30. x=v cosh(1u), y=v sinh(1u), z=u+w2
- Answer
- vu2
31. x=u+v, y=v+w, z=u
32. x=u−v, y=u+v, z=u+v+w
- Answer
- 2
33. The triangular region R with the vertices (0,0), (1,1), and (1,2) is shown in the following figure.
a. Find a transformation T:S→R, T(u,v)=(x,y)=(au+bv+dv), where a,b,c, and d are real numbers with ad−bc≠0 such that T−1(0,0)=(0,0), T−1(1,1)=(1,0), and T−1(1,2)=(0,1).
b. Use the transformation T to find the area A(R) of the region R.
34. The triangular region R with the vertices (0,0), (2,0), and (1,3) is shown in the following figure.
a. Find a transformation T:S→R, T(u,v)=(x,y)=(au+bv+dv), where a,b,c, and d are real numbers with ad−bc≠0 such that T−1(0,0)=(0,0), T−1(2,0)=(1,0), and T−1(1,3)=(0,1).
b. Use the transformation T to find the area A(R) of the region R.
- Answer
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a. T(u,v)=(2u+v, 3v)
b. The area of R is A(R)=∫30∫(6−y)/3y/3dxdy=∫10∫1−u0|∂(x,y)∂(u,v)|dv du=∫10∫1−u06dvdu=3.
In exercises 35 - 36, use the transformation u=y−x, v=y, to evaluate the integrals on the parallelogram R of vertices (0,0), (1,0), (2,1), and (1,1) shown in the following figure.
35. ∬
36. \displaystyle \iint_R (y^2 - xy) \, dA
- Answer
- -\frac{1}{4}
In exercises 37 - 38, use the transformation y = x = u, x + y = v to evaluate the integrals on the square R determined by the lines y = x, y = -x + 2, y = x + 2, and y = -x shown in the following figure.
37. \displaystyle \iint_R e^{x+y} \, dA
38. \displaystyle \iint_R \sin (x - y) \, dA
- Answer
- -1 + \cos 2
In exercises 39 - 40, use the transformation x = u, 5y = v to evaluate the integrals on the region R bounded by the ellipse x^2 + 25y^2 = 1 shown in the following figure.
39. \displaystyle \iint_R \sqrt{x^2 + 25y^2} \, dA
40. \displaystyle \iint_R (x^2 + 25y^2)^2 \, dA
- Answer
- \frac{\pi}{15}
In exercises 41 - 42, use the transformation u = x + y, v = x - y to evaluate the integrals on the trapezoidal region R determined by the points (1,0), (2,0), (0,2), and (0,1) shown in the following figure.
41. \displaystyle \iint_R (x^2 - 2xy + y^2) \space e^{x+y} \, dA
42. \displaystyle \iint_R (x^3 + 3x^2y + 3xy^2 + y^3) \, dA
- Answer
- \frac{31}{5}
43. The circular annulus sector R bounded by the circles 4x^2 + 4y^2 = 1 and 9x^2 + 9y^2 = 64, the line x = y \sqrt{3}, and the y-axis is shown in the following figure. Find a transformation T from a rectangular region S in the r\theta-plane to the region R in the xy-plane. Graph S.
44. The solid R bounded by the circular cylinder x^2 + y^2 = 9 and the planes z = 0, \space z = 1, \space x = 0, and y = 0 is shown in the following figure. Find a transformation T from a cylindrical box S in r\theta z-space to the solid R in xyz-space.
- Answer
- T (r,\theta,z) = (r \space \cos \theta, \space r \space \sin \theta, \space z); \space S = [0,3] \times [0,\frac{\pi}{2}] \times [0,1] in the r\theta z-space
45. Show that \iint_R f \left(\sqrt{\frac{x^2}{3} + \frac{y^2}{3}}\right) dA = 2 \pi \sqrt{15} \int_0^1 f (\rho) \rho \space d\rho, \nonumber where f is a continuous function on [0,1] and R is the region bounded by the ellipse 5x^2 + 3y^2 = 15.
46. Show that \iiint_R f \left(\sqrt{16x^2 + 4y^2 + z^2}\right) dV = \frac{\pi}{2} \int_0^1 f (\rho) \rho^2 d\rho, \nonumber where f is a continuous function on [0,1] and R is the region bounded by the ellipsoid 16x^2 + 4y^2 + z^2 = 1.
47. [T] Find the area of the region bounded by the curves xy = 1, \space xy = 3, \space y = 2x, and y = 3x by using the transformation u = xy and v = \frac{y}{x}. Use a computer algebra system (CAS) to graph the boundary curves of the region R.
48. [T] Find the area of the region bounded by the curves x^2y = 2, \space x^2y = 3, \space y = x, and y = 2x by using the transformation u = x^2y and v = \frac{y}{x}. Use a CAS to graph the boundary curves of the region R.
- Answer
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The area of R is 10 - 4\sqrt{6}; the boundary curves of R are graphed in the following figure.
49. Evaluate the triple integral \int_0^1 \int_1^2 \int_z^{z+1} (y + 1) \space dx \space dy \space dz \nonumber by using the transformation u = x - z, \space v = 3y, and w = \frac{z}{2}.
50. Evaluate the triple integral \int_0^2 \int_4^6 \int_{3z}^{3z+2} (5 - 4y) \space dx \space dy \space dz \nonumber by using the transformation u = x - 3z, \space v = 4y, and w = z.
- Answer
- 8
51. A transformation T : R^2 \rightarrow R^2, \space T (u,v) = (x,y) of the form x = au + bv, \space y = cu + dv, where a,b,c, and d are real numbers, is called linear. Show that a linear transformation for which ad - bc \neq 0 maps parallelograms to parallelograms.
52. A transformation T_{\theta} : R^2 \rightarrow R^2, \space T_{\theta} (u,v) = (x,y) of the form x = u \space \cos \theta - v \space \sin \theta, \space y = u \space \sin \theta + v \space \cos \theta, is called a rotation angle \theta. Show that the inverse transformation of T_{\theta} satisfies T_{\theta}^{-1} = T_{-\theta} where T_{-\theta} is the rotation of angle -\theta.
53. [T] Find the region S in the uv-plane whose image through a rotation of angle \frac{\pi}{4} is the region R enclosed by the ellipse x^2 + 4y^2 = 1. Use a CAS to answer the following questions.
a. Graph the region S.
b. Evaluate the integral \displaystyle \iint_S e^{-2uv} \, du \, dv. Round your answer to two decimal places.
54. [T] The transformations T_i : \mathbb{R}^2 \rightarrow \mathbb{R}^2, \space i = 1, . . . , 4, defined by T_1(u,v) = (u,-v), \space T_2 (u,v) = (-u,v), \space T_3 (u,v) = (-u, -v), and T_4 (u,v) = (v,u) are called reflections about the x-axis, y-axis origin, and the line y = x, respectively.
a. Find the image of the region S = \big\{(u,v)\,|\,u^2 + v^2 - 2u - 4v + 1 \leq 0\big\} in the xy-plane through the transformation T_1 \circ T_2 \circ T_3 \circ T_4.
b. Use a CAS to graph R.
c. Evaluate the integral \displaystyle \iint_S \sin (u^2) \, du \, dv by using a CAS. Round your answer to two decimal places.
- Answer
-
a. R = \big\{(x,y)\,|\,y^2 + x^2 - 2y - 4x + 1 \leq 0\big\};
b. R is graphed in the following figure;c. 3.16
55. [T] The transformations T_{k,1,1} : \mathbb{R}^3 \rightarrow \mathbb{R}^3, \space T_{k,1,1}(u,v,w) = (x,y,z) of the form x = ku, \space y = v, \space z = w, where k \neq 1 is a positive real number, is called a stretch if k > 1 and a compression if 0 < k < 1 in the x-direction. Use a CAS to evaluate the integral \displaystyle \iiint_S e^{-(4x^2+9y^2+25z^2)} \, dx \, dy \, dz on the solid S = \big\{(x,y,z) \,|\, 4x^2 + 9y^2 + 25z^2 \leq 1\big\} by considering the compression T_{2,3,5}(u,v,w) = (x,y,z) defined by x = \frac{u}{2}, \space y = \frac{v}{3}, and z = \frac{w}{5}. Round your answer to four decimal places.
56. [T] The transformation T_{a,0} : \mathbb{R}^2 \rightarrow \mathbb{R}^2, \space T_{a,0} (u,v) = (u + av, v), where a \neq 0 is a real number, is called a shear in the x-direction. The transformation, T_{b,0} : R^2 \rightarrow R^2, \space T_{o,b}(u,v) = (u,bu + v), where b \neq 0 is a real number, is called a shear in the y-direction.
a. Find transformations T_{0,2} \circ T_{3,0}.
b. Find the image R of the trapezoidal region S bounded by u = 0, \space v = 0, \space v = 1, and v = 2 - u through the transformation T_{0,2} \circ T_{3,0}.
c. Use a CAS to graph the image R in the xy-plane.
d. Find the area of the region R by using the area of region S.
- Answer
-
a. T_{0,2} \circ T_{3,0}(u,v) = (u + 3v, 2u + 7v);
b. The image S is the quadrilateral of vertices (0,0), \space (3,7), \space (2,4), and (4,9);
c. S is graphed in the following figure;
d. \frac{3}{2}
57. Use the transformation, x = au, \space y = av, \space z = cw and spherical coordinates to show that the volume of a region bounded by the spheroid \frac{x^2+y^2}{a^2} + \frac{z^2}{c^2} = 1 is \frac{4\pi a^2c}{3}.
58. Find the volume of a football whose shape is a spheroid \frac{x^2+y^2}{a^2} + \frac{z^2}{c^2} = 1 whose length from tip to tip is 11 inches and circumference at the center is 22 inches. Round your answer to two decimal places.
- Answer
- \frac{2662}{3\pi} \approx 282.45 \space in^3
Contributors
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.