16.5E: Ejercicios para la Sección 16.5
- Page ID
- 116742
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Para los siguientes ejercicios, determine si la declaración es Verdadero o Falso.
1. Si las funciones de coordenadas de\(\vecs F : \mathbb{R}^3 \rightarrow \mathbb{R}^3\) tienen segundas derivadas parciales continuas, entonces\(\text{curl} \, (\text{div} \,\vecs F)\) es igual a cero.
2. \(\vecs\nabla \cdot (x \,\mathbf{\hat i} + y \,\mathbf{\hat j} + z \,\mathbf{\hat k} ) = 1\).
- Contestar
- Falso
3. Todos los campos vectoriales de la forma\(\vecs F(x,y,z) = f(x)\,\mathbf{\hat i} + g(y)\,\mathbf{\hat j} + h(z)\,\mathbf{\hat k}\) son conservadores.
4. Si\(\text{curl} \, \vecs F = \vecs 0\), entonces\(\vecs F\) es conservador.
- Contestar
- Cierto
5. Si\(\vecs F\) es un campo vector constante entonces\(\text{div} \,\vecs F = 0\).
6. Si\(\vecs F\) es un campo vector constante entonces\(\text{curl} \,\vecs F =\vecs 0\).
- Contestar
- Cierto
Para los siguientes ejercicios, encuentra el rizo de\(\vecs F\).
7. \(\vecs F(x,y,z) = xy^2z^4\,\mathbf{\hat i} + (2x^2y + z)\,\mathbf{\hat j} + y^3 z^2\,\mathbf{\hat k}\)
8. \(\vecs F(x,y,z) = x^2 z\,\mathbf{\hat i} + y^2 x\,\mathbf{\hat j} + (y + 2z)\,\mathbf{\hat k}\)
- Contestar
- \(\text{curl} \,\vecs F = \,\mathbf{\hat i} + x^2\,\mathbf{\hat j} + y^2\,\mathbf{\hat k}\)
9. \(\vecs F(x,y,z) = 3xyz^2\,\mathbf{\hat i} + y^2 \sin z\,\mathbf{\hat j} + xe^{2z}\,\mathbf{\hat k}\)
10. \(\vecs F(x,y,z) = x^2 yz\,\mathbf{\hat i} + xy^2 z\,\mathbf{\hat j} + xyz^2\,\mathbf{\hat k}\)
- Contestar
- \(\text{curl} \, \vecs F = (xz^2 - xy^2)\,\mathbf{\hat i} + (x^2 y - yz^2)\,\mathbf{\hat j} + (y^2z - x^2z)\,\mathbf{\hat k}\)
11. \(\vecs F(x,y,z) = (x \, \cos y)\,\mathbf{\hat i} + xy^2\,\mathbf{\hat j}\)
12. \(\vecs F(x,y,z) = (x - y)\,\mathbf{\hat i} + (y - z)\,\mathbf{\hat j} + (z - x)\,\mathbf{\hat k}\)
- Contestar
- \(\text{curl }\, \vecs F = \,\mathbf{\hat i} + \,\mathbf{\hat j} + \,\mathbf{\hat k}\)
13. \(\vecs F(x,y,z) = xyz\,\mathbf{\hat i} + x^2y^2z^2 \,\mathbf{\hat j} + y^2z^3 \,\mathbf{\hat k}\)
14. \(\vecs F(x,y,z) = xy\,\mathbf{\hat i} + yz \,\mathbf{\hat j} + xz \,\mathbf{\hat k}\)
- Contestar
- \(\text{curl }\, \vecs F = - y\,\mathbf{\hat i} - z \,\mathbf{\hat j} - x \,\mathbf{\hat k}\)
15. \(\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + y^2 \,\mathbf{\hat j} + z^2 \,\mathbf{\hat k}\)
16. \(\vecs F(x,y,z) = ax\,\mathbf{\hat i} + by \,\mathbf{\hat j} + c \,\mathbf{\hat k}\)para constantes\(a, \,b, \,c\).
- Contestar
- \(\text{curl }\, \vecs F = \vecs 0\)
Para los siguientes ejercicios, encuentra la divergencia de\(\vecs F\).
17. \(\vecs F(x,y,z) = x^2 z\,\mathbf{\hat i} + y^2 x \,\mathbf{\hat j} + (y + 2z) \,\mathbf{\hat k}\)
18. \(\vecs F(x,y,z) = 3xyz^2\,\mathbf{\hat i} + y^2 \sin z \,\mathbf{\hat j} + xe^2 \,\mathbf{\hat k}\)
- Contestar
- \(\text{div}\,\vecs F = 3yz^2 + 2y \, \sin z + 2xe^{2z}\)
19. \(\vecs{F}(x,y) = (\sin x)\,\mathbf{\hat i} + (\cos y) \,\mathbf{\hat j}\)
20. \(\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + y^2 \,\mathbf{\hat j} + z^2 \,\mathbf{\hat k}\)
- Contestar
- \(\text{div}\,\vecs F = 2(x + y + z)\)
21. \(\vecs F(x,y,z) = (x - y)\,\mathbf{\hat i} + (y - z) \,\mathbf{\hat j} + (z - x) \,\mathbf{\hat k}\)
22. \(\vecs{F}(x,y) = \dfrac{x}{\sqrt{x^2+y^2}}\,\mathbf{\hat i} + \dfrac{y}{\sqrt{x^2+y^2}}\,\mathbf{\hat j}\)
- Contestar
- \(\text{div}\,\vecs F = \dfrac{1}{\sqrt{x^2+y^2}}\)
23. \(\vecs{F}(x,y) = x\,\mathbf{\hat i} - y \,\mathbf{\hat j}\)
24. \(\vecs F(x,y,z) = ax\,\mathbf{\hat i} + by \,\mathbf{\hat j} + c \,\mathbf{\hat k}\)para constantes\(a, \,b, \,c\).
- Contestar
- \(\text{div}\,\vecs F = a + b\)
25. \(\vecs F(x,y,z) = xyz\,\mathbf{\hat i} + x^2y^2z^2\,\mathbf{\hat j} + y^2z^3\,\mathbf{\hat k}\)
26. \(\vecs F(x,y,z) = xy\,\mathbf{\hat i} + yz\,\mathbf{\hat j} + xz\,\mathbf{\hat k}\)
- Contestar
- \(\text{div}\,\vecs F = x + y + z\)
Para los ejercicios 27 y 28, determine si cada una de las funciones escalares dadas es armónica.
27. \(u(x,y,z) = e^{-x} (\cos y - \sin y)\)
28. \(w(x,y,z) = (x^2 + y^2 + z^2)^{-1/2}\)
- Contestar
- Armónico
29. Si\(\vecs F(x,y,z) = 2\,\mathbf{\hat i} + 2x j + 3y k\) y\(\vecs G(x,y,z) = x\,\mathbf{\hat i} - y \,\mathbf{\hat j} + z \,\mathbf{\hat k}\), encuentra\(\text{curl} \, (\vecs F \times \vecs G)\).
30. Si\(\vecs F(x,y,z) = 2\,\mathbf{\hat i} + 2x j + 3y k\) y\(\vecs G(x,y,z) = x\,\mathbf{\hat i} - y \,\mathbf{\hat j} + z \,\mathbf{\hat k}\), encuentra\(\text{div} \, (\vecs F \times \vecs G)\).
- Contestar
- \(\text{div} \, (\vecs F \times \vecs G) = 2z + 3x\)
31. Encontrar\(\text{div} \,\vecs F\), dado eso\(\vecs F = \vecs \nabla f\), dónde\(f(x,y,z) = xy^3z^2\).
32. Encuentra la divergencia de\(\vecs F\) para campo vectorial\(\vecs F(x,y,z) = (y^2 + z^2) (x + y) \,\mathbf{\hat i} + (z^2 + x^2)(y + z) \,\mathbf{\hat j} + (x^2 + y^2)(z + x) \,\mathbf{\hat k}\).
- Contestar
- \(\text{div}\,\vecs F = 2r^2\)
33. Encuentra la divergencia de\(\vecs F\) para campo vectorial\(\vecs F(x,y,z) = f_1(y,z)\,\mathbf{\hat i} + f_2 (x,z) \,\mathbf{\hat j} + f_3 (x,y) \,\mathbf{\hat k}\).
Para ejercicios 34 - 36, use\(r = |\vecs r|\) y\(\vecs r(x,y,z) = \langle x,y,z\rangle\).
34. Encuentra el\(\text{curl} \, \vecs r\)
- Contestar
- \(\text{curl} \, \vecs r = \vecs 0\)
35. Encuentra el\(\text{curl}\, \dfrac{\vecs r}{r}\).
36. Encuentra el\(\text{curl}\, \dfrac{\vecs r}{r^3}\).
- Contestar
- \(\text{curl}\, \dfrac{\vecs r}{r^3} = \vecs 0\)
37. Let\(\vecs{F}(x,y) = \dfrac{-y\,\mathbf{\hat i}+x\,\mathbf{\hat j}}{x^2+y^2}\), donde\(\vecs F\) se define en\(\big\{(x,y) \in \mathbb{R} | (x,y) \neq (0,0) \big\}\). Encuentra\(\text{curl}\, \vecs F\).
Para los siguientes ejercicios, utilice un sistema de álgebra por computadora para encontrar el rizo de los campos vectoriales dados.
38. [T]\(\vecs F(x,y,z) = \arctan \left(\dfrac{x}{y}\right)\,\mathbf{\hat i} + \ln \sqrt{x^2 + y^2} \,\mathbf{\hat j}+ \,\mathbf{\hat k}\)
- Contestar
- \(\text{curl }\, \vecs F = \dfrac{2x}{x^2+y^2}\,\mathbf{\hat k}\)
39. [T]\(\vecs F(x,y,z) = \sin (x - y)\,\mathbf{\hat i} + \sin (y - z) \,\mathbf{\hat j} + \sin (z - x) \,\mathbf{\hat k}\)
Para los siguientes ejercicios, encuentra la divergencia de\(\vecs F\) en el punto dado.
40. \(\vecs F(x,y,z) = \,\mathbf{\hat i} + \,\mathbf{\hat j} + \,\mathbf{\hat k}\)en\((2, -1, 3)\)
- Contestar
- \(\text{div}\,\vecs F = 0\)
41. \(\vecs F(x,y,z) = xyz \,\mathbf{\hat i} + y \,\mathbf{\hat j} + z\,\mathbf{\hat k}\)en\((1, 2, 3)\)
42. \(\vecs F(x,y,z) = e^{-xy}\,\mathbf{\hat i} + e^{xz}\,\mathbf{\hat j} + e^{yz}\,\mathbf{\hat k}\)en\((3, 2, 0)\)
- Contestar
- \(\text{div}\,\vecs F = 2 - 2e^{-6}\)
43. \(\vecs F(x,y,z) = xyz \,\mathbf{\hat i} + y \,\mathbf{\hat j} + z\,\mathbf{\hat k}\)en\((1, 2, 1)\)
44. \(\vecs F(x,y,z) = e^x \sin y \,\mathbf{\hat i} - e^x \cos y\,\mathbf{\hat j} \)en\((0, 0, 3)\)
- Contestar
- \(\text{div}\,\vecs F = 0\)
Para los ejercicios 45- 49, encuentra el rizo de\(\vecs F\) en el punto dado.
45. \(\vecs F(x,y,z) = \,\mathbf{\hat i} + \,\mathbf{\hat j} + \,\mathbf{\hat k}\)en\((2, -1, 3)\)
46. \(\vecs F(x,y,z) = xyz \,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}\)en\((1, 2, 3)\)
- Contestar
- \(\text{curl }\, \vecs F = \mathbf{\hat j} - 3\,\mathbf{\hat k}\)
47. \(\vecs F(x,y,z) = e^{-xy}\,\mathbf{\hat i} + e^{xz}\,\mathbf{\hat j} + e^{yz}\,\mathbf{\hat k}\)en\((3, 2, 0)\)
48. \(\vecs F(x,y,z) = xyz \,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}\)en\((1, 2, 1)\)
- Contestar
- \(\text{curl }\, \vecs F = 2\,\mathbf{\hat j} - \,\mathbf{\hat k}\)
49. \(\vecs F(x,y,z) = e^x \sin y \,\mathbf{\hat i} - e^x \cos y\,\mathbf{\hat j} \)en\((0, 0, 3)\)
50. Vamos\(\vecs F(x,y,z) = (3x^2 y + az) \,\mathbf{\hat i} + x^3\,\mathbf{\hat j} + (3x + 3z^2)\,\mathbf{\hat k}\). ¿Para qué valor de\(a\) es\(\vecs F\) conservador?
- Contestar
- \(a = 3\)
51. Dado el campo vectorial\(\vecs{F}(x,y) = \dfrac{1}{x^2+y^2} \langle -y,x\rangle\) en el dominio\(D = \dfrac{\mathbb{R}^2}{\{(0,0)\}} = \big\{(x,y) \in \mathbb{R}^2 |(x,y) \neq (0,0) \big\}\), ¿es\(\vecs F\) conservador?
52. Dado el campo vectorial\(\vecs{F}(x,y) = \dfrac{1}{x^2+y^2} \langle x,y\rangle\) en el dominio\(D = \dfrac{\mathbb{R}^2}{\{(0,0)\}}\), ¿es\(\vecs F\) conservador?
- Contestar
- \(\vecs F\)es conservadora.
53. Encuentra el trabajo realizado por campo\(\vecs{F}(x,y) = e^{-y}\,\mathbf{\hat i} - xe^{-y}\,\mathbf{\hat j}\) de fuerza al mover un objeto de\(P(0, 1)\) a\(Q(2, 0)\). ¿El campo de fuerza es conservador?
54. Divergencia de\(\vecs F(x,y,z) = (\sinh x)\,\mathbf{\hat i} + (\cosh y)\,\mathbf{\hat j} - xyz\,\mathbf{\hat k}\) cómputos
- Contestar
- \(\text{div}\,\vecs F = \cosh x + \sinh y - xy\)
55. Cómputos\(\text{curl }\, \vecs F = (\sinh x)\,\mathbf{\hat i} + (\cosh y)\,\mathbf{\hat j} - xyz\,\mathbf{\hat k}\).
Para los siguientes ejercicios, considere un cuerpo rígido que esté girando alrededor del\(x\) eje en sentido antihorario con velocidad angular constante\(\vecs \omega = \langle a,b,c \rangle\). Si\(P\) es un punto en el cuerpo ubicado en\(\vecs r = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}\), la velocidad a\(P\) viene dada por el campo vectorial\(\vecs F = \vecs \omega \times \vecs r\).
. The object is roughly a sphere with pointed ends on the x axis, which cuts it in half. An arrow r is drawn from (0,0,0) to P(x,y,z) and down from P(x,y,z) to the x axis." data-type="media" id="fs-id1167793480282">. El objeto es aproximadamente una esfera con extremos puntiagudos en el eje x, que lo corta por la mitad. Se dibuja una flecha r de (0,0,0) a P (x, y, z) y hacia abajo desde P (x, y, z) hasta el eje x." src="https://math.libretexts.org/@api/dek...16_05_201.jpeg">
56. Express\(\vecs F\) in terms of \(\,\mathbf{\hat i},\;\,\mathbf{\hat j},\) and \(\,\mathbf{\hat k}\) vectors.
- Answer
- \(\vecs F = (bz - cy)\,\mathbf{\hat i}+(cx - az)\,\mathbf{\hat j} + (ay - bx)\,\mathbf{\hat k}\)
57. Find \(\text{div} \, F\).
58. Find \(\text{curl} \, F\)
- Answer
- \(\text{curl }\, \vecs F = 2\vecs\omega\)
In the following exercises, suppose that \(\vecs \nabla \cdot \vecs F = 0\) and \(\vecs \nabla \cdot \vecs G = 0\).
59. Does \(\vecs F + \vecs G\) necessarily have zero divergence?
60. Does \(\vecs F \times \vecs G\) necessarily have zero divergence?
- Answer
- \(\vecs F \times \vecs G\) does not have zero divergence.
In the following exercises, suppose a solid object in \(\mathbb{R}^3\) has a temperature distribution given by \(T(x,y,z)\). The heat flow vector field in the object is \(\vecs F = - k \vecs \nabla T\), where \(k > 0\) is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\vecs \nabla \cdot \vecs F = -k \vecs \nabla \cdot \vecs \nabla T = - k \vecs \nabla^2 T\).
61. Compute the heat flow vector field.
62. Compute the divergence.
- Answer
- \(\vecs \nabla \cdot \vecs F = -200 k [1 + 2(x^2 + y^2 + z^2)] e^{-x^2+y^2+z^2}\)
63. [T] Consider rotational velocity field \(\vecs v = \langle 0,10z, -10y \rangle\). If a paddlewheel is placed in plane \(x + y + z = 1\) with its axis normal to this plane, using a computer algebra system, calculate how fast the paddlewheel spins in revolutions per unit time.
