2.8E: Exercises

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

Answer

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.

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

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

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

Answer

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.

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

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

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

Answer

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 $(0,2)$.

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. $x = 2u, \space y = 3v$, 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 $(1,-1)$.

Answer: $T$ 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 $S = R = R^3$.

Answer: $T$ 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 $S = R = R^2$.

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

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 $R = R^2$.

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

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 $S = R = R^3$.

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

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 $a,b > 0$

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

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 $a,b,c > 0$

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

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

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

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

Answer: $\frac{3}{2}$

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

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

Answer: $-1$

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

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

Answer: $2uv$

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

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

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

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

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

Answer: $2$

33. The triangular region $R$ with the vertices $(0,0), \space (1,1)$, 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 $T : S \rightarrow R, \space T(u,v) = (x,y) = (au + bv + dv)$, 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.

$A triangle with corners at the origin, (2, 0), and (1, 3).$

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

Answer: 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 $u = y - x$, $v = y$, 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. $\displaystyle \iint_R (y - x) \, dA$

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.

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

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.

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

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.

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

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 $S$.

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

44. The solid $R$ bounded by the circular cylinder $x^2 + y^2 = 9$ 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.

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

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

The area of $R$ is $10 - 4\sqrt{6}$; 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 \[\int_0^1 \int_1^2 \int_z^{z+1} (y + 1) \space dx \space dy \space dz \nonumber \] 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;

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

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;

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

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.

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