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Mathematics LibreTexts

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

    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).

    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 \(R\).

    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 \(u = y - x, \space v = y\), 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.

    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 \(y = x = u, \space x + y = v\) 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.

    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 \(x = u, \space 5y = v\) to evaluate the integrals on the region \(R\) bounded by the ellipse \(x^2 + 25y^2 = 1\) shown in the following figure.

    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 \(u = x + y, \space v = x - y\) 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.

    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 \[\iint_R f \left(\sqrt{\frac{x^2}{3} + \frac{y^2}{3}}\right) dA = 2 \pi \sqrt{15} \int_0^1 f (\rho) \rho \space d\rho,\] 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 \[\iiint_R f \left(\sqrt{16x^2 + 4y^2 + z^2}\right) dV = \frac{\pi}{2} \int_0^1 f (\rho) \rho^2 d\rho,\] 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 \[\int_0^1 \int_1^2 \int_z^{z+1} (y + 1) \space dx \space dy \space dz\] by using the transformation \(u = x - z, \space v = 3y\), and \(w = \frac{z}{2}\).

    50. Evaluate the triple integral \[\int_0^2 \int_4^6 \int_{3z}^{3z+2} (5 - 4y) \space dx \space dy \space dz\] 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.