# 14.7E: Exercises for Section 14.7

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In exercises 1 - 6, the function $$T : S \rightarrow R, \space T (u,v) = (x,y)$$ on the region $$S = \big\{(u,v) \,|\, 0 \leq u \leq 1, \space 0 \leq v \leq 1\big\}$$ bounded by the unit square is given, where $$R \in R^2$$ is the image of $$S$$ under $$T$$.

a. Justify that the function $$T$$ is a $$C^1$$ 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. $$x = 2u, \space y = 3v$$

2. $$x = \frac{u}{2}, \space y = \frac{v}{3}$$

a. $$T(u,v) = (g(u,v), \space h(u,v), \space x = g(u,v) = \frac{u}{2}$$ and $$y = h(u,v) = \frac{v}{3}$$. The functions $$g$$ and $$h$$ are continuous and differentiable, and the partial derivatives $$g_u (u,v) = \frac{1}{2}, \space g_v (u,v) = 0, \space h_u (u,v) = 0$$ and $$h_v (u,v) = \frac{1}{3}$$ are continuous on $$S$$;

b. $$T(0,0) = (0,0), \space T(1,0) = \left(\frac{1}{2},0\right), \space T(0,1) = \left(0,\frac{1}{3}\right)$$, and $$T(1,1) = \left(\frac{1}{2}, \frac{1}{3} \right)$$;

c. $$R$$ is the rectangle of vertices $$(0,0), \space \left(0,\frac{1}{3}\right), \space \left(\frac{1}{2}, \frac{1}{3} \right)$$, and $$\left(0,\frac{1}{3}\right)$$ in the $$xy$$-plane; the following figure.

3. $$x = u - v, \space y = u + v$$

4. $$x = 2u - v, \space y = u + 2v$$

a. $$T(u,v) = (g(u,v), \space h(u,v), \space x = g(u,v) = 2u - v$$ and $$y = h(u,v) = u + 2v$$. The functions $$g$$ and $$h$$ are continuous and differentiable, and the partial derivatives $$g_u (u,v) = 2, \space g_v (u,v) = -1, \space h_u (u,v) = 1$$ and $$h_v (u,v) = 2$$ are continuous on $$S$$;

b. $$T(0,0) = (0,0), \space T(1,0) = (2,1), \space T(0,1) = (-1,2)$$, and $$T(1,1) = (1,3)$$;

c. $$R$$ is the parallelogram of vertices $$(0,0), \space (2,1) \space (1,3)$$, and $$(-1,2)$$ in the $$xy$$-plane; the following figure.

5. $$x = u^2, \space y = v^2$$

6. $$x = u^3, \space y = v^3$$

a. $$T(u,v) = (g(u,v), \space h(u,v), \space x = g(u,v) = u^3$$ and $$y = h(u,v) = v^3$$. The functions $$g$$ and $$h$$ are continuous and differentiable, and the partial derivatives $$g_u (u,v) = 3u^2, \space g_v (u,v) = 0, \space h_u (u,v) = 0$$ and $$h_v (u,v) = 3v^2$$ are continuous on $$S$$;

b. $$T(0,0) = (0,0), \space T(1,0) = (1,0), \space T(0,1) = (0,1)$$, 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 $$T : S \rightarrow R$$ are one-to-one or not.

7. $$x = u^2, \space y = v^2$$, where $$S$$ is the rectangle of vertices $$(-1,0), \space (1,0), \space (1,1)$$, and $$(-1,1)$$.

8. $$x = u^4, \space y = u^2 + v$$, where $$S$$ is the triangle of vertices $$(-2,0), \space (2,0)$$, and $$(0,2)$$.

$$T$$ is not one-to-one: two points of $$S$$ have the same image. Indeed, $$T(-2,0) = T(2,0) = (16,4)$$.

9. $$x = 2u, \space y = 3v$$, where $$S$$ is the square of vertices $$(-1,1), \space (-1,-1), \space (1,-1)$$, and $$(1,1)$$.

10. $$T(u, v) = (2u - v, u),$$ where $$S$$ is the triangle with vertices $$(-1,1), \, (-1,-1)$$, and $$(1,-1)$$.

$$T$$ is one-to-one: We argue by contradiction. $$T(u_1,v_1) = T(u_2,v_2)$$ implies $$2u_1 - v_1 = 2u_2 - v_2$$ and $$u_1 = u_2$$. Thus, $$u_1 = u+2$$ and $$v_1 = v_2$$.

11. $$x = u + v + w, \space y = u + v, \space z = w$$, where $$S = R = R^3$$.

12. $$x = u^2 + v + w, \space y = u^2 + v, \space z = w$$, where $$S = R = R^3$$.

$$T$$ is not one-to-one: $$T(1,v,w) = (-1,v,w)$$

In exercises 13 - 18, the transformations $$T : R \rightarrow S$$ are one-to-one. Find their related inverse transformations $$T^{-1} : R \rightarrow S$$.

13. $$x = 4u, \space y = 5v$$, where $$S = R = R^2$$.

14. $$x = u + 2v, \space y = -u + v$$, where $$S = R = R^2$$.

$$u = \frac{x-2y}{3}, \space v= \frac{x+y}{3}$$

15. $$x = e^{2u+v}, \space y = e^{u-v}$$, where $$S = R^2$$ and $$R = \big\{(x,y) \,|\, x > 0, \space y > 0\big\}$$

16. $$x = \ln u, \space y = \ln(uv)$$, where $$S = \big\{(u,v) \,|\, u > 0, \space v > 0\big\}$$ and $$R = R^2$$.

$$u = e^x, \space v = e^{-x+y}$$

17. $$x = u + v + w, \space y = 3v, \space z = 2w$$, where $$S = R = R^3$$.

18. $$x = u + v, \space y = v + w, \space z = u + w$$, where $$S = R = R^3$$.

$$u = \frac{x-y+z}{2}, \space v = \frac{x+y-z}{2}, \space w = \frac{-x+y+z}{2}$$

In exercises 19 - 22, the transformation $$T : S \rightarrow R, \space T (u,v) = (x,y)$$ and the region $$R \subset R^2$$ are given. Find the region $$S \subset R^2$$.

19. $$x = au, \space y = bv, \space R = \big\{(x,y) \,|\, x^2 + y^2 \leq a^2 b^2\big\}$$ where $$a,b > 0$$

20. $$x = au, \space y = bc, \space R = \big\{(x,y) \,|\, \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1\big\}$$, where $$a,b > 0$$

$$S = \big\{(u,v) \,|\, u^2 + v^2 \leq 1\big\}$$

21. $$x = \frac{u}{a}, \space y = \frac{v}{b}, \space z = \frac{w}{c}, \space R = \big\{(x,y)\,|\,x^2 + y^2 + z^2 \leq 1\big\}$$, where $$a,b,c > 0$$

22. $$x = au, \space y = bv, \space z = cw, \space R = \big\{(x,y)\,|\,\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} \leq 1, \space z > 0\big\}$$, where $$a,b,c > 0$$

$$R = \big\{(u,v,w)\,|\,u^2 - v^2 - w^2 \leq 1, \space w > 0\big\}$$

In exercises 23 - 32, find the Jacobian $$J$$ of the transformation.

23. $$x = u + 2v, \space y = -u + v$$

24. $$x = \frac{u^3}{2}, \space y = \frac{v}{u^2}$$

$$\frac{3}{2}$$

25. $$x = e^{2u-v}, \space y = e^{u+v}$$

26. $$x = ue^v, \space y = e^{-v}$$

$$-1$$

27. $$x = u \space \cos (e^v), \space y = u \space \sin(e^v)$$

28. $$x = v \space \sin (u^2), \space y = v \space \cos(u^2)$$

$$2uv$$

29. $$x = u \space \cosh v, \space y = u \space \sinh v, \space z = w$$

30. $$x = v \space \cosh \left(\frac{1}{u}\right), \space y = v \space \sinh \left(\frac{1}{u}\right), \space z = u + w^2$$

$$\frac{v}{u^2}$$

31. $$x = u + v, \space y = v + w, \space z = u$$

32. $$x = u - v, \space y = u + v, \space z = u + v + w$$

$$2$$

33. The triangular region $$R$$ with the vertices $$(0,0), \space (1,1)$$, and $$(1,2)$$ is shown in the following figure.

a. Find a transformation $$T : S \rightarrow R, \space T(u,v) = (x,y) = (au + bv + dv)$$, where $$a,b,c$$, and $$d$$ are real numbers with $$ad - bc \neq 0$$ such that $$T^{-1} (0,0) = (0,0), \space T^{-1} (1,1) = (1,0)$$, 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 $$(0,0), \space (2,0)$$, and $$(1,3)$$ is shown in the following figure.

a. Find a transformation $$T : S \rightarrow R, \space T(u,v) = (x,y) = (au + bv + dv)$$, where $$a,b,c$$, and $$d$$ are real numbers with $$ad - bc \neq 0$$ such that $$T^{-1} (0,0) = (0,0), \space T^{-1} (2,0) = (1,0)$$, and $$T^{-1}(1,3) = (0,1)$$.

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

a. $$T (u,v) = (2u + v, \space 3v)$$
b. The area of $$R$$ is $$\displaystyle A(R) = \int_0^3 \int_{y/3}^{(6-y)/3} \, dx \, dy = \int_0^1 \int_0^{1-u} \left|\frac{\partial (x,y)}{\partial (u,v)}\right| \, dv \space du = \int_0^1 \int_0^{1-u} 6 \, dv \, du = 3.$$

In exercises 35 - 36, use the transformation $$u = y - x, \space v = y$$, to evaluate the integrals on the parallelogram $$R$$ of vertices $$(0,0), \space (1,0), \space (2,1)$$, and $$(1,1)$$ shown in the following figure.

35. $$\displaystyle \iint_R (y - x) \, dA$$

36. $$\displaystyle \iint_R (y^2 - xy) \, dA$$

$$-\frac{1}{4}$$

In exercises 37 - 38, use the transformation $$y = x = u, \space x + y = v$$ to evaluate the integrals on the square $$R$$ determined by the lines $$y = x, \space y = -x + 2, \space y = x + 2$$, and $$y = -x$$ shown in the following figure.

37. $$\displaystyle \iint_R e^{x+y} \, dA$$

38. $$\displaystyle \iint_R \sin (x - y) \, dA$$

$$-1 + \cos 2$$

In exercises 39 - 40, use the transformation $$x = u, \space 5y = v$$ 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. $$\displaystyle \iint_R (x^2 + 25y^2)^2 \, dA$$

$$\frac{\pi}{15}$$

In exercises 41 - 42, use the transformation $$u = x + y, \space v = x - y$$ to evaluate the integrals on the trapezoidal region $$R$$ determined by the points $$(1,0), \space (2,0), \space (0,2)$$, and $$(0,1)$$ shown in the following figure.

41. $$\displaystyle \iint_R (x^2 - 2xy + y^2) \space e^{x+y} \, dA$$

42. $$\displaystyle \iint_R (x^3 + 3x^2y + 3xy^2 + y^3) \, dA$$

$$\frac{31}{5}$$

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.

$$T (r,\theta,z) = (r \space \cos \theta, \space r \space \sin \theta, \space z); \space S = [0,3] \times [0,\frac{\pi}{2}] \times [0,1]$$ 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 $$R$$.

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 $$w = z$$.

$$8$$

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.

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 $$S$$.

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.

$$\frac{2662}{3\pi} \approx 282.45 \space in^3$$