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16.8E: Ejercicios para la Sección 16.8

• Edwin “Jed” Herman & Gilbert Strang
• OpenStax

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Para los ejercicios 1 - 9, use un sistema algebraico por computadora (CAS) y el teorema de divergencia para evaluar la integral de superficie$$\displaystyle \int_S \vecs F \cdot \vecs n \, ds$$ para la elección dada de$$\vecs F$$ y la superficie límite$$S.$$ Para cada superficie cerrada, asumir$$\vecs N$$ es el vector normal de unidad exterior.

1. [T]$$\vecs F(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}$$;$$S$$ es la superficie del cubo$$0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 < z \leq 1$$.

2. [T]$$\vecs F(x,y,z) = (\cos yz) \,\mathbf{\hat i} + e^{xz}\,\mathbf{\hat j} + 3z^2 \,\mathbf{\hat k}$$;$$S$$ es la superficie del hemisferio$$z = \sqrt{4 - x^2 - y^2}$$ junto con el disco$$x^2 + y^2 \leq 4$$ en el$$xy$$ plano.

Contestar
$$\displaystyle \int_S \vecs F \cdot \vecs n \, ds = 75.3982$$

3. [T]$$\vecs F(x,y,z) = (x^2 + y^2 - x^2)\,\mathbf{\hat i} + x^2 y\,\mathbf{\hat j} + 3z\,\mathbf{\hat k};$$$$S$$ es la superficie de las cinco caras del cubo unitario$$0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 < z \leq 1.$$

4. [T]$$\vecs F(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k};$$$$S$$ es la superficie de paraboloide$$z = x^2 + y^2$$ para$$0 \leq z \leq 9$$.

Contestar
$$\displaystyle \int_S \vecs F \cdot \vecs n \, ds = 127.2345$$

5. [T]$$\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + y^2 \,\mathbf{\hat j} + z^2 \,\mathbf{\hat k}$$;$$S$$ es la superficie de la esfera$$x^2 + y^2 + z^2 = 4$$.

6. [T]$$\vecs F(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + (z^2 - 1)\,\mathbf{\hat k}$$;$$S$$ es la superficie del sólido delimitada por cilindro$$x^2 + y^2 = 4$$ y planos$$z = 0$$ y$$z = 1$$.

Contestar
$$\displaystyle \int_S \vecs F \cdot \vecs n \, ds = 37.699$$

7. [T]$$\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + y^2 \,\mathbf{\hat j} + z^2 \,\mathbf{\hat k}$$;$$S$$ es la superficie delimitada arriba por esfera$$\rho = 2$$ y abajo por cono$$\varphi = \dfrac{\pi}{4}$$ en coordenadas esféricas. (Piense en$$S$$ como la superficie de un “cono de helado”.)

8. [T]$$\vecs F(x,y,z) = x^3\,\mathbf{\hat i} + y^3 \,\mathbf{\hat j} + 3a^2z \,\mathbf{\hat k} \, (constant \, a > 0)$$;$$S$$ es la superficie delimitada por cilindro$$x^2 + y^2 = a^2$$ y planos$$z = 0$$ y$$z = 1$$.

Contestar
$$\displaystyle \int_S \vecs F \cdot \vecs n \, ds = \dfrac{9\pi a^4}{2}$$

9. [T] Integral de superficie$$\displaystyle \iint_S \vecs F \cdot dS$$, donde$$S$$ está el sólido limitado por paraboloide$$z = x^2 + y^2$$ y plano$$z = 4$$, y$$\vecs F(x,y,z) = (x + y^2z^2)\,\mathbf{\hat i} + (y + z^2x^2)\,\mathbf{\hat j} + (z + x^2y^2) \,\mathbf{\hat k}$$

10. Utilice el teorema de divergencia para calcular la integral de superficie$$\displaystyle \iint_S \vecs F \cdot dS$$, donde$$\vecs F(x,y,z) = (e^{y^2} \,\mathbf{\hat i} + (y + \sin (z^2))\,\mathbf{\hat j} + (z - 1)\,\mathbf{\hat k}$$ y$$S$$ es el hemisferio superior$$x^2 + y^2 + z^2 = 1, \, z \geq 0$$, orientada hacia arriba.

Contestar
$$\displaystyle \iint_S \vecs F \cdot dS = \dfrac{\pi}{3}$$

11. Utilice el teorema de divergencia para calcular la integral de superficie$$\displaystyle \iint_S \vecs F \cdot dS$$, donde$$\vecs F(x,y,z) = x^4\,\mathbf{\hat i} - x^3z^2\,\mathbf{\hat j} + 4xy^2z\,\mathbf{\hat k}$$ y$$S$$ es la superficie delimitada por cilindro$$x^2 + y^2 = 1$$ y planos$$z = x + 2$$ y$$z = 0$$.

12. Utilice el teorema de divergencia para calcular la integral de superficie$$\displaystyle \iint_S \vecs F \cdot dS$$, cuando$$\vecs F(x,y,z) = x^2z^3 \,\mathbf{\hat i} + 2xyz^3\,\mathbf{\hat j} + xz^4 \,\mathbf{\hat k}$$ y$$S$$ es la superficie de la caja con vértices$$(\pm 1, \, \pm 2, \, \pm 3)$$.

Contestar
$$\displaystyle \iint_S \vecs F \cdot dS = 0$$

13. Utilizar el teorema de divergencia para calcular la integral de superficie$$\displaystyle \iint_S \vecs F \cdot dS$$, cuando$$\vecs F(x,y,z) = z \, \tan^{-1} (y^2)\,\mathbf{\hat i} + z^3 \ln(x^2 + 1) \,\mathbf{\hat j} + z\,\mathbf{\hat k}$$ y$$S$$ es una parte del paraboloide$$x^2 + y^2 + z = 2$$ que se encuentra por encima del plano$$z = 1$$ y se orienta hacia arriba.

14. [T] Usar un CAS y el teorema de divergencia para calcular el flujo$$\displaystyle \iint_S \vecs F \cdot dS$$, donde$$\vecs F(x,y,z) = (x^3 + y^3)\,\mathbf{\hat i} + (y^3 + z^3)\,\mathbf{\hat j} + (z^3 + x^3)\,\mathbf{\hat k}$$ y$$S$$ es una esfera con centro$$(0, 0)$$ y radio$$2.$$

Contestar
$$\displaystyle \iint_S \vecs F \cdot dS = 241.2743$$

15. Utilice el teorema de divergencia para calcular el valor de la integral de flujo$$\displaystyle \iint_S \vecs F \cdot dS$$, donde$$\vecs F(x,y,z) = (y^3 + 3x)\,\mathbf{\hat i} + (xz + y)\,\mathbf{\hat j} + \left(z + x^4 \cos (x^2y)\right)\,\mathbf{\hat k}$$ y$$S$$ es el área de la región delimitada por$$x^2 + y^2 = 1, \, x \geq 0, \, y \geq 0$$, y$$0 \leq z \leq 1$$.

0, y>0, and z>0. A quarter of a cylinder is drawn with center on the z axis. The arrows have positive x, y, and z components; they point away from the origin." data-type="media"> 0, y>0 y z>0. Un cuarto de cilindro se dibuja con el centro en el eje z. Las flechas tienen componentes x, y y z positivos; apuntan lejos del origen." src="https://math.libretexts.org/@api/dek...16_08_202.jpeg">

16. Usar el teorema de divergencia para calcular la integral de flujo$$\displaystyle \iint_S \vecs F \cdot dS$$, where $$\vecs F(x,y,z) = y\,\mathbf{\hat j} - z\,\mathbf{\hat k}$$ and $$S$$ consists of the union of paraboloid $$y = x^2 + z^2, \, 0 \leq y \leq 1$$, and disk $$x^2 + z^2 \leq 1, \, y = 1$$, oriented outward. What is the flux through just the paraboloid?

$$\displaystyle \iint_S \vecs F \cdot dS = -\pi$$

17. Use the divergence theorem to compute flux integral $$\displaystyle \iint_S \vecs F \cdot dS$$, where $$\vecs F(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z^4 \,\mathbf{\hat k}$$ and $$S$$ is a part of cone $$z = \sqrt{x^2 + y^2}$$ beneath top plane $$z = 1$$ oriented downward.

18. Use the divergence theorem to calculate surface integral $$\displaystyle \iint_S \vecs F \cdot dS$$ for $$\vecs F(x,y,z) = x^4\,\mathbf{\hat i} - x^3z^2\,\mathbf{\hat j} + 4xy^2 z\,\mathbf{\hat k}$$, where $$S$$ is the surface bounded by cylinder $$x^2 + y^2 = 1$$ and planes $$z = x + 2$$ and $$z = 0$$.

$$\displaystyle \iint_S \vecs F \cdot dS = \dfrac{2\pi}{3}$$

19. Consider $$\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + xy\,\mathbf{\hat j} + (z + 1)\,\mathbf{\hat k}$$. Let $$E$$ be the solid enclosed by paraboloid $$z = 4 - x^2 - y^2$$ and plane $$z = 0$$ with normal vectors pointing outside $$E.$$ Compute flux $$\vecs F$$ across the boundary of $$E$$ using the divergence theorem.

In exercises 20 - 23, use a CAS along with the divergence theorem to compute the net outward flux for the fields across the given surfaces $$S.$$

20. [T] $$\vecs F = \langle x,\, -2y, \, 3z \rangle;$$ $$S$$ is sphere $$\{(x,y,z) : x^2 + y^2 + z^2 = 6 \}$$.

$$15\sqrt{6}\pi$$

21. [T] $$\vecs F = \langle x, \, 2y, \, z \rangle$$; $$S$$ is the boundary of the tetrahedron in the first octant formed by plane $$x + y + z = 1$$.

22. [T] $$\vecs F = \langle y - 2x, \, x^3 - y, \, y^2 - z \rangle$$; $$S$$ is sphere $$\{(x,y,z) \,:\, x^2 + y^2 + z^2 = 4\}.$$

$$-\dfrac{128}{3} \pi$$

23. [T] $$\vecs F = \langle x,y,z \rangle$$; $$S$$ is the surface of paraboloid $$z = 4 - x^2 - y^2$$, for $$z \geq 0$$, plus its base in the $$xy$$-plane.

For exercises 24 - 26, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions $$D.$$

24. [T] $$\vecs F = \langle z - x, \, x - y, \, 2y - z \rangle$$; $$D$$ is the region between spheres of radius 2 and 4 centered at the origin.

$$-703.7168$$

25. [T] $$\vecs F = \dfrac{\vecs r}{\|\vecs r\|} = \dfrac{\langle x,y,z\rangle}{\sqrt{x^2+y^2+z^2}}$$; $$D$$ is the region between spheres of radius 1 and 2 centered at the origin.

26. [T] $$\vecs F = \langle x^2, \, -y^2, \, z^2 \rangle$$; $$D$$ is the region in the first octant between planes $$z = 4 - x - y$$ and $$z = 2 - x - y$$.

$$20$$

27. Let $$\vecs F(x,y,z) = 2x\,\mathbf{\hat i} - 3xy\,\mathbf{\hat j} + xz^2\,\mathbf{\hat k}$$. Use the divergence theorem to calculate $$\displaystyle \iint_S \vecs F \cdot dS$$, where $$S$$ is the surface of the cube with corners at $$(0,0,0), \, (1,0,0), \, (0,1,0), \, (1,1,0), \, (0,0,1), \, (1,0,1), \, (0,1,1)$$, and $$(1,1,1)$$, oriented outward.

28. Use the divergence theorem to find the outward flux of field $$\vecs F(x,y,z) = (x^3 - 3y)\,\mathbf{\hat i} + (2yz + 1)\,\mathbf{\hat j} + xyz\,\mathbf{\hat k}$$ through the cube bounded by planes $$x = \pm 1, \, y = \pm 1,$$ and $$z = \pm 1$$.

$$\displaystyle \iint_S \vecs F \cdot dS = 8$$

29. Let $$\vecs F(x,y,z) = 2x\,\mathbf{\hat i} - 3y\,\mathbf{\hat j} + 5z\,\mathbf{\hat k}$$ and let $$S$$ be hemisphere $$z = \sqrt{9 - x^2 - y^2}$$ together with disk $$x^2 + y^2 \leq 9$$ in the $$xy$$-plane. Use the divergence theorem.

30. Evaluate $$\displaystyle \iint_S \vecs F \cdot \vecs n \, dS$$, where $$\vecs F(x,y,z) = x^2 \,\mathbf{\hat i} + xy\,\mathbf{\hat j} + x^3y^3\,\mathbf{\hat k}$$ and $$S$$ is the surface consisting of all faces except the tetrahedron bounded by plane $$x + y + z = 1$$ and the coordinate planes, with outward unit normal vector $$\vecs N.$$

Contestar
$$\displaystyle \iint_S \vecs F \cdot \vecs n \, dS = \dfrac{1}{8}$$

31. Encuentre el flujo neto de campo hacia afuera$$\vecs F = \langle bz - cy, \, cx - az, \, ay - bx \rangle$$ a través de cualquier superficie lisa y cerrada en$$R^3$$ donde$$a, \, b,$$ y$$c$$ son constantes.

32. Utilizar el teorema de divergencia para evaluar$$\displaystyle \iint_S ||\vecs R||\vecs R \cdot \vecs n \, ds,$$ dónde$$\vecs R(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}$$ y$$S$$ es esfera$$x^2 + y^2 + z^2 = a^2$$, con constante$$a > 0$$.

Contestar
$$\displaystyle \iint_S ||\vecs R||\vecs R \cdot \vecs n \, ds = 4\pi a^4$$

33. Utilice el teorema de divergencia para evaluar$$\displaystyle \iint_S \vecs F \cdot dS,$$ dónde$$\vecs F(x,y,z) = y^2 z\,\mathbf{\hat i} + y^3\,\mathbf{\hat j} + xz\,\mathbf{\hat k}$$ y$$S$$ es el límite del cubo definido por$$-1 \leq x \leq 1, \, -1 \leq y \leq 1$$, y$$0 \leq z \leq 2$$.

34. Dejar$$R$$ ser la región definida por$$x^2 + y^2 + z^2 \leq 1$$. Usa el teorema de la divergencia para encontrar$$\displaystyle \iiint_R z^2 \, dV.$$

Contestar
$$\displaystyle \iiint_R z^2 dV = \dfrac{4\pi}{15}$$

35. Dejar$$E$$ ser el sólido delimitado por el$$xy$$ -plano y paraboloide de$$z = 4 - x^2 - y^2$$ manera que$$S$$ sea la superficie de la pieza paraboloide junto con el disco en el$$xy$$ -plano que forma su fondo. Si$$\vecs F(x,y,z) = (xz \, \sin(yz) + x^3) \,\mathbf{\hat i} + \cos (yz) \,\mathbf{\hat j} + (3zy^2 - e^{x^2+y^2})\,\mathbf{\hat k}$$, encontrar$$\displaystyle \iint_S \vecs F \cdot dS$$ usando el teorema de la divergencia.

36. Let

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