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2.8E: Exercises

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In exercises 1 - 6, the function T:SR, T(u,v)=(x,y) on the region S={(u,v)|0u1, 0v1} bounded by the unit square is given, where RR2 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

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.

A rectangle with one corner at the origin, horizontal length 0.5, and vertical height 0.34.

3. x=uv, y=u+v

4. x=2uv, y=u+2v

Answer

a. T(u,v)=(g(u,v), h(u,v), x=g(u,v)=2uv 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.

A square of side length square root of 5 with one corner at the origin and another at (2, 1).

5. x=u2, y=v2

6. x=u3, y=v3

Answer

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:SR 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)=(2uv,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 2u1v1=2u2v2 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:RS are one-to-one. Find their related inverse transformations T1:RS.

13. x=4u, y=5v, where S=R=R2.

14. x=u+2v, y=u+v, where S=R=R2.

Answer
u=x2y3, v=x+y3

15. x=e2u+v, y=euv, 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=ex+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=xy+z2, v=x+yz2, w=x+y+z2

In exercises 19 - 22, the transformation T:SR, T(u,v)=(x,y) and the region RR2 are given. Find the region SR2.

19. x=au, y=bv, R={(x,y)|x2+y2a2b2} where a,b>0

20. x=au, y=bc, R={(x,y)|x2a2+y2b21}, where a,b>0

Answer
S={(u,v)|u2+v21}

21. x=ua, y=vb, z=wc, R={(x,y)|x2+y2+z21}, where a,b,c>0

22. x=au, y=bv, z=cw, R={(x,y)|x2a2y2b2z2c21, z>0}, where a,b,c>0

Answer
R={(u,v,w)|u2v2w21, 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=e2uv, y=eu+v

26. x=uev, y=ev

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=uv, 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 triangle with corners at the origin, (1, 1), and (1, 2).

a. Find a transformation T:SR, T(u,v)=(x,y)=(au+bv+dv), where a,b,c, and d are real numbers with adbc0 such that T1(0,0)=(0,0), T1(1,1)=(1,0), and T1(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 triangle with corners at the origin, (2, 0), and (1, 3).

a. Find a transformation T:SR, T(u,v)=(x,y)=(au+bv+dv), where a,b,c, and d are real numbers with adbc0 such that T1(0,0)=(0,0), T1(2,0)=(1,0), and T1(1,3)=(0,1).

b. Use the transformation T to find the area A(R) of the region R.

Answer

a. T(u,v)=(2u+v, 3v)
b. The area of R is A(R)=30(6y)/3y/3dxdy=101u0|(x,y)(u,v)|dv du=101u06dvdu=3.

In exercises 35 - 36, use the transformation u=yx, 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.

A rhombus with corners at the origin, (1, 0), (1, 1), and (2, 1).

35. R(yx)dA

36. R(y2xy)dA

Answer
14

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.

A square with side lengths square root of 2 rotated 45 degrees with one corner at the origin and another at (1, 1).

37. Rex+ydA

38. Rsin(xy)dA

Answer
1+cos2

In exercises 39 - 40, use the transformation x=u, 5y=v to evaluate the integrals on the region R bounded by the ellipse x2+25y2=1 shown in the following figure.

An ellipse with center at the origin, major axis 2, and minor 0.4.

39. Rx2+25y2dA

40. R(x2+25y2)2dA

Answer
π15

In exercises 41 - 42, use the transformation u=x+y, v=xy 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.

A trapezoid with corners at (1, 0), (0, 1), (0, 2), and (2, 0).

41. R(x22xy+y2) ex+ydA

42. R(x3+3x2y+3xy2+y3)dA

Answer
315

43. The circular annulus sector R bounded by the circles 4x2+4y2=1 and 9x2+9y2=64, the line x=y3, and the y-axis is shown in the following figure. Find a transformation T from a rectangular region S in the rθ-plane to the region R in the xy-plane. Graph S.

In the first quadrant, a section of an annulus described by an inner radius of 0.5, outer radius slightly more than 2.5, and center the origin. There is a line dividing this annulus that comes from approximately a 30 degree angle. The portion corresponding to 60 degrees is shaded.

44. The solid R bounded by the circular cylinder x2+y2=9 and the planes z=0, z=1, x=0, and y=0 is shown in the following figure. Find a transformation T from a cylindrical box S in rθz-space to the solid R in xyz-space.

A quarter of a cylinder with height 1 and radius 3. The center axis is the z axis.

Answer
T(r,θ,z)=(r cosθ, r sinθ, z); S=[0,3]×[0,π2]×[0,1] in the rθz-space

45. Show that Rf(x23+y23)dA=2π1510f(ρ)ρ dρ, where f is a continuous function on [0,1] and R is the region bounded by the ellipse 5x2+3y2=15.

46. Show that Rf(16x2+4y2+z2)dV=π210f(ρ)ρ2dρ, where f is a continuous function on [0,1] and R is the region bounded by the ellipsoid 16x2+4y2+z2=1.

47. [T] Find the area of the region bounded by the curves xy=1, xy=3, y=2x, and y=3x by using the transformation u=xy and v=yx. 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 x2y=2, x2y=3, y=x, and y=2x by using the transformation u=x2y and v=yx. Use a CAS to graph the boundary curves of the region R.

Answer

The area of R is 1046; the boundary curves of R are graphed in the following figure.

Four lines are drawn, namely, y = 3, y = 2, y = 3/(x squared), and y = 2/(x squared). The lines y = 3 and y = 2 are parallel to each other. The lines y = 3/(x squared) and y = 2/(x squared) are curves that run somewhat parallel to each other.

49. Evaluate the triple integral 1021z+1z(y+1) dx dy dz by using the transformation u=xz, v=3y, and w=z2.

50. Evaluate the triple integral 20643z+23z(54y) dx dy dz by using the transformation u=x3z, v=4y, and w=z.

Answer
8

51. A transformation T:R2R2, T(u,v)=(x,y) of the form x=au+bv, y=cu+dv, where a,b,c, and d are real numbers, is called linear. Show that a linear transformation for which adbc0 maps parallelograms to parallelograms.

52. A transformation Tθ:R2R2, Tθ(u,v)=(x,y) of the form x=u cosθv sinθ, y=u sinθ+v cosθ, is called a rotation angle θ. Show that the inverse transformation of Tθ satisfies T1θ=Tθ where Tθ is the rotation of angle θ.

53. [T] Find the region S in the uv-plane whose image through a rotation of angle π4 is the region R enclosed by the ellipse x2+4y2=1. Use a CAS to answer the following questions.

a. Graph the region S.

b. Evaluate the integral Se2uvdudv. Round your answer to two decimal places.

54. [T] The transformations Ti:R2R2, i=1,...,4, defined by T1(u,v)=(u,v), T2(u,v)=(u,v), T3(u,v)=(u,v), and T4(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={(u,v)|u2+v22u4v+10} in the xy-plane through the transformation T1T2T3T4.

b. Use a CAS to graph R.

c. Evaluate the integral Ssin(u2)dudv by using a CAS. Round your answer to two decimal places.

Answer

a. R={(x,y)|y2+x22y4x+10};
b. R is graphed in the following figure;

A circle with radius 2 and center (2, 1).

c. 3.16

55. [T] The transformations Tk,1,1:R3R3, Tk,1,1(u,v,w)=(x,y,z) of the form x=ku, y=v, z=w, where k1 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 Se(4x2+9y2+25z2)dxdydz on the solid S={(x,y,z)|4x2+9y2+25z21} by considering the compression T2,3,5(u,v,w)=(x,y,z) defined by x=u2, y=v3, and z=w5. Round your answer to four decimal places.

56. [T] The transformation Ta,0:R2R2, Ta,0(u,v)=(u+av,v), where a0 is a real number, is called a shear in the x-direction. The transformation, Tb,0:R2R2, To,b(u,v)=(u,bu+v), where b0 is a real number, is called a shear in the y-direction.

a. Find transformations T0,2T3,0.

b. Find the image R of the trapezoidal region S bounded by u=0, v=0, v=1, and v=2u through the transformation T0,2T3,0.

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. T0,2T3,0(u,v)=(u+3v,2u+7v);

b. The image S is the quadrilateral of vertices (0,0), (3,7), (2,4), and (4,9);

c. S is graphed in the following figure;

A four-sided figure with points the origin, (2, 4), (4, 9), and (3, 7).

d. 32

57. Use the transformation, x=au, y=av, z=cw and spherical coordinates to show that the volume of a region bounded by the spheroid x2+y2a2+z2c2=1 is 4πa2c3.

58. Find the volume of a football whose shape is a spheroid x2+y2a2+z2c2=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
26623π282.45 in3

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.


2.8E: Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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