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16.5E: Exercícios para a Seção 16.5

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    188538

    • Edwin “Jed” Herman & Gilbert Strang
    • OpenStax
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    Para os exercícios a seguir, determine se a afirmação é verdadeira ou falsa.

    1. Se as funções de coordenadas de\(\vecs F : \mathbb{R}^3 \rightarrow \mathbb{R}^3\) tiverem derivadas parciais secundárias contínuas, então\(\text{curl} \, (\text{div} \,\vecs F)\) é igual a zero.

    2. \(\vecs\nabla \cdot (x \,\mathbf{\hat i} + y \,\mathbf{\hat j} + z \,\mathbf{\hat k} ) = 1\).

    Responda
    Falso

    3. Todos os campos vetoriais do formulário\(\vecs F(x,y,z) = f(x)\,\mathbf{\hat i} + g(y)\,\mathbf{\hat j} + h(z)\,\mathbf{\hat k}\) são conservadores.

    4. Se\(\text{curl} \, \vecs F = \vecs 0\), então\(\vecs F\) é conservador.

    Responda
    É verdade

    5. Se\(\vecs F\) for um campo vetorial constante, então\(\text{div} \,\vecs F = 0\).

    6. Se\(\vecs F\) for um campo vetorial constante, então\(\text{curl} \,\vecs F =\vecs 0\).

    Responda
    É verdade

    Para os exercícios a seguir, encontre a curvatura 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. \(\vecs F(x,y,z) = x^2 z\,\mathbf{\hat i} + y^2 x\,\mathbf{\hat j} + (y + 2z)\,\mathbf{\hat k}\)

    Responda
    \(\text{curl} \,\vecs F = \,\mathbf{\hat i} + x^2\,\mathbf{\hat j} + y^2\,\mathbf{\hat k}\)

    9. \(\vecs F(x,y,z) = 3xyz^2\,\mathbf{\hat i} + y^2 \sin z\,\mathbf{\hat j} + xe^{2z}\,\mathbf{\hat k}\)

    10. \(\vecs F(x,y,z) = x^2 yz\,\mathbf{\hat i} + xy^2 z\,\mathbf{\hat j} + xyz^2\,\mathbf{\hat k}\)

    Responda
    \(\text{curl} \, \vecs F = (xz^2 - xy^2)\,\mathbf{\hat i} + (x^2 y - yz^2)\,\mathbf{\hat j} + (y^2z - x^2z)\,\mathbf{\hat k}\)

    11. \(\vecs F(x,y,z) = (x \, \cos y)\,\mathbf{\hat i} + xy^2\,\mathbf{\hat j}\)

    12. \(\vecs F(x,y,z) = (x - y)\,\mathbf{\hat i} + (y - z)\,\mathbf{\hat j} + (z - x)\,\mathbf{\hat k}\)

    Responda
    \(\text{curl }\, \vecs F = \,\mathbf{\hat i} + \,\mathbf{\hat j} + \,\mathbf{\hat k}\)

    13. \(\vecs F(x,y,z) = xyz\,\mathbf{\hat i} + x^2y^2z^2 \,\mathbf{\hat j} + y^2z^3 \,\mathbf{\hat k}\)

    14. \(\vecs F(x,y,z) = xy\,\mathbf{\hat i} + yz \,\mathbf{\hat j} + xz \,\mathbf{\hat k}\)

    Responda
    \(\text{curl }\, \vecs F = - y\,\mathbf{\hat i} - z \,\mathbf{\hat j} - x \,\mathbf{\hat k}\)

    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\(a, \,b, \,c\).

    Responda
    \(\text{curl }\, \vecs F = \vecs 0\)

    Para os exercícios a seguir, encontre a divergência 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. \(\vecs F(x,y,z) = 3xyz^2\,\mathbf{\hat i} + y^2 \sin z \,\mathbf{\hat j} + xe^2 \,\mathbf{\hat k}\)

    Responda
    \(\text{div}\,\vecs F = 3yz^2 + 2y \, \sin z + 2xe^{2z}\)

    19. \(\vecs{F}(x,y) = (\sin x)\,\mathbf{\hat i} + (\cos y) \,\mathbf{\hat j}\)

    20. \(\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + y^2 \,\mathbf{\hat j} + z^2 \,\mathbf{\hat k}\)

    Responda
    \(\text{div}\,\vecs F = 2(x + y + z)\)

    21. \(\vecs F(x,y,z) = (x - y)\,\mathbf{\hat i} + (y - z) \,\mathbf{\hat j} + (z - x) \,\mathbf{\hat k}\)

    22. \(\vecs{F}(x,y) = \dfrac{x}{\sqrt{x^2+y^2}}\,\mathbf{\hat i} + \dfrac{y}{\sqrt{x^2+y^2}}\,\mathbf{\hat j}\)

    Responda
    \(\text{div}\,\vecs F = \dfrac{1}{\sqrt{x^2+y^2}}\)

    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\(a, \,b, \,c\).

    Responda
    \(\text{div}\,\vecs F = a + b\)

    25. \(\vecs F(x,y,z) = xyz\,\mathbf{\hat i} + x^2y^2z^2\,\mathbf{\hat j} + y^2z^3\,\mathbf{\hat k}\)

    26. \(\vecs F(x,y,z) = xy\,\mathbf{\hat i} + yz\,\mathbf{\hat j} + xz\,\mathbf{\hat k}\)

    Responda
    \(\text{div}\,\vecs F = x + y + z\)

    Para os exercícios 27 e 28, determine se cada uma das funções escalares dadas é harmônica.

    27. \(u(x,y,z) = e^{-x} (\cos y - \sin y)\)

    28. \(w(x,y,z) = (x^2 + y^2 + z^2)^{-1/2}\)

    Responda
    Harmônico

    29. Se\(\vecs F(x,y,z) = 2\,\mathbf{\hat i} + 2x j + 3y k\) e\(\vecs G(x,y,z) = x\,\mathbf{\hat i} - y \,\mathbf{\hat j} + z \,\mathbf{\hat k}\), encontre\(\text{curl} \, (\vecs F \times \vecs G)\).

    30. Se\(\vecs F(x,y,z) = 2\,\mathbf{\hat i} + 2x j + 3y k\) e\(\vecs G(x,y,z) = x\,\mathbf{\hat i} - y \,\mathbf{\hat j} + z \,\mathbf{\hat k}\), encontre\(\text{div} \, (\vecs F \times \vecs G)\).

    Responda
    \(\text{div} \, (\vecs F \times \vecs G) = 2z + 3x\)

    31. Descubra\(\text{div} \,\vecs F\), considerando isso\(\vecs F = \vecs \nabla f\), onde\(f(x,y,z) = xy^3z^2\).

    32. Encontre a divergência de\(\vecs F\) para o campo vetorial\(\vecs F(x,y,z) = (y^2 + z^2) (x + y) \,\mathbf{\hat i} + (z^2 + x^2)(y + z) \,\mathbf{\hat j} + (x^2 + y^2)(z + x) \,\mathbf{\hat k}\).

    Responda
    \(\text{div}\,\vecs F = 2r^2\)

    33. Encontre a divergência de\(\vecs F\) para o campo vetorial\(\vecs F(x,y,z) = f_1(y,z)\,\mathbf{\hat i} + f_2 (x,z) \,\mathbf{\hat j} + f_3 (x,y) \,\mathbf{\hat k}\).

    Para os exercícios 34 a 36, use\(r = |\vecs r|\)\(\vecs r(x,y,z) = \langle x,y,z\rangle\) e.

    34. Encontre o\(\text{curl} \, \vecs r\)

    Responda
    \(\text{curl} \, \vecs r = \vecs 0\)

    35. Encontre\(\text{curl}\, \dfrac{\vecs r}{r}\) o.

    36. Encontre\(\text{curl}\, \dfrac{\vecs r}{r^3}\) o.

    Responda
    \(\text{curl}\, \dfrac{\vecs r}{r^3} = \vecs 0\)

    37. Let\(\vecs{F}(x,y) = \dfrac{-y\,\mathbf{\hat i}+x\,\mathbf{\hat j}}{x^2+y^2}\), onde\(\vecs F\) está definido em\(\big\{(x,y) \in \mathbb{R} | (x,y) \neq (0,0) \big\}\). Encontre\(\text{curl}\, \vecs F\).

    Para os exercícios a seguir, use um sistema computacional de álgebra para encontrar a curvatura dos campos vetoriais fornecidos.

    38. [T]\(\vecs F(x,y,z) = \arctan \left(\dfrac{x}{y}\right)\,\mathbf{\hat i} + \ln \sqrt{x^2 + y^2} \,\mathbf{\hat j}+ \,\mathbf{\hat k}\)

    Responda
    \(\text{curl }\, \vecs F = \dfrac{2x}{x^2+y^2}\,\mathbf{\hat k}\)

    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 os exercícios a seguir, encontre a divergência de\(\vecs F\) no ponto determinado.

    40. \(\vecs F(x,y,z) = \,\mathbf{\hat i} + \,\mathbf{\hat j} + \,\mathbf{\hat k}\)em\((2, -1, 3)\)

    Responda
    \(\text{div}\,\vecs F = 0\)

    41. \(\vecs F(x,y,z) = xyz \,\mathbf{\hat i} + y \,\mathbf{\hat j} + z\,\mathbf{\hat k}\)em\((1, 2, 3)\)

    42. \(\vecs F(x,y,z) = e^{-xy}\,\mathbf{\hat i} + e^{xz}\,\mathbf{\hat j} + e^{yz}\,\mathbf{\hat k}\)em\((3, 2, 0)\)

    Responda
    \(\text{div}\,\vecs F = 2 - 2e^{-6}\)

    43. \(\vecs F(x,y,z) = xyz \,\mathbf{\hat i} + y \,\mathbf{\hat j} + z\,\mathbf{\hat k}\)em\((1, 2, 1)\)

    44. \(\vecs F(x,y,z) = e^x \sin y \,\mathbf{\hat i} - e^x \cos y\,\mathbf{\hat j} \)em\((0, 0, 3)\)

    Responda
    \(\text{div}\,\vecs F = 0\)

    Para os exercícios 45- 49, encontre a curvatura de\(\vecs F\) no ponto determinado.

    45. \(\vecs F(x,y,z) = \,\mathbf{\hat i} + \,\mathbf{\hat j} + \,\mathbf{\hat k}\)em\((2, -1, 3)\)

    46. \(\vecs F(x,y,z) = xyz \,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}\)em\((1, 2, 3)\)

    Responda
    \(\text{curl }\, \vecs F = \mathbf{\hat j} - 3\,\mathbf{\hat k}\)

    47. \(\vecs F(x,y,z) = e^{-xy}\,\mathbf{\hat i} + e^{xz}\,\mathbf{\hat j} + e^{yz}\,\mathbf{\hat k}\)em\((3, 2, 0)\)

    48. \(\vecs F(x,y,z) = xyz \,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}\)em\((1, 2, 1)\)

    Responda
    \(\text{curl }\, \vecs F = 2\,\mathbf{\hat j} - \,\mathbf{\hat k}\)

    49. \(\vecs F(x,y,z) = e^x \sin y \,\mathbf{\hat i} - e^x \cos y\,\mathbf{\hat j} \)em\((0, 0, 3)\)

    50. Deixe\(\vecs F(x,y,z) = (3x^2 y + az) \,\mathbf{\hat i} + x^3\,\mathbf{\hat j} + (3x + 3z^2)\,\mathbf{\hat k}\). Por que valor de\(a\) é\(\vecs F\) conservador?

    Responda
    \(a = 3\)

    51. Dado o campo vetorial\(\vecs{F}(x,y) = \dfrac{1}{x^2+y^2} \langle -y,x\rangle\) no domínio\(D = \dfrac{\mathbb{R}^2}{\{(0,0)\}} = \big\{(x,y) \in \mathbb{R}^2 |(x,y) \neq (0,0) \big\}\), é\(\vecs F\) conservador?

    52. Dado o campo vetorial\(\vecs{F}(x,y) = \dfrac{1}{x^2+y^2} \langle x,y\rangle\) no domínio\(D = \dfrac{\mathbb{R}^2}{\{(0,0)\}}\), é\(\vecs F\) conservador?

    Responda
    \(\vecs F\)é conservador.

    53. Encontre o trabalho realizado pelo campo\(\vecs{F}(x,y) = e^{-y}\,\mathbf{\hat i} - xe^{-y}\,\mathbf{\hat j}\) de força ao mover um objeto de\(P(0, 1)\) para\(Q(2, 0)\). O campo de força é conservador?

    54. Divergência computacional\(\vecs F(x,y,z) = (\sinh x)\,\mathbf{\hat i} + (\cosh y)\,\mathbf{\hat j} - xyz\,\mathbf{\hat k}\).

    Responda
    \(\text{div}\,\vecs F = \cosh x + \sinh y - xy\)

    55. Computar\(\text{curl }\, \vecs F = (\sinh x)\,\mathbf{\hat i} + (\cosh y)\,\mathbf{\hat j} - xyz\,\mathbf{\hat k}\).

    Para os exercícios a seguir, considere um corpo rígido que gira em torno do\(x\) eixo -no sentido anti-horário com velocidade angular constante\(\vecs \omega = \langle a,b,c \rangle\). Se\(P\) for um ponto no corpo localizado em\(\vecs r = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}\), a velocidade em\(P\) é dada pelo campo vetorial\(\vecs F = \vecs \omega \times \vecs r\).

    . 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">Um diagrama tridimensional de um objeto girando em torno do eixo x no sentido anti-horário com velocidade angular constante w = <a, b, c. O objeto é aproximadamente uma esfera com pontas pontiagudas no eixo x, que o corta ao meio. Uma seta r é desenhada de (0,0,0) para P (x, y, z) e para baixo de P (x, y, z) até o eixo x." src="https://math.libretexts.org/@api/dek...16_05_201.jpeg">

    56. Expressa\(\vecs F\) 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.

    A three dimensional diagram of a rotational velocity field. The arrows are showing a rotation in a clockwise manner. A paddlewheel is shown in plan x + y + z = 1 with n extended out perpendicular to the plane.