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True or False The divergence of a vector field \(\mathbf{ F}\) is a vector.

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If \(\mathbf{F} (x,y,z)=P(x,y,z) \mathbf{i}+ Q(x,y,z)\mathbf{j}+R(x,y,z)\mathbf{k}\) is a vector field, where \(\dfrac{\partial P}{\partial x}\), \(\dfrac{\partial Q}{ \partial y}\), and \(\dfrac{\partial R}{\partial z}\) each exist, then \(\rm{ div}\mathbf{F}=\)__________.

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

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\(\mathbf{F}(x,y,z)=x\mathbf{i}+xy\mathbf{j}+xyz\mathbf{k}\)

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\(\mathbf{F}(x,y,z)=(x+\cos x)\mathbf{i}+(y+y\sin x)\mathbf{j}+2z\mathbf{k}\)

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\(\mathbf{F}(x,y,z)=xye^{z}\mathbf{i}+x^{2}e^{z}\mathbf{j}+x^{2}ye^{z}\mathbf{k}\)

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\(\mathbf{F}(x,y,z)= \sqrt{x^2+y^2} \mathbf{i}+ \sqrt{x^2+y^2} \mathbf{j} +z\mathbf{k} \)

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\(\mathbf{F}(x,y,z) = xy^2 \mathbf{i} + x^2y \mathbf{j} + xyz{\bf k}\)

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\(\mathbf{F}=(2xy+2z)\mathbf{i}+(y^{2}+1)\mathbf{j} -(x+y)\mathbf{k}\); \(S\) is the surface of the solid enclosed by \(x+y+z=4\), \(x=0\), \(y=0\), and \(z=0\).

Question

\(\mathbf{F}=(2xy+z)\mathbf{i}+y^{2}\mathbf{j}-(x+4y)\mathbf{k}\) ; \(S\) is the surface of the solid enclosed by \(2x+2y+z=6\), \(x=0\), \(y=0\), and \(z=0\).

Question

\(\mathbf{F}=x^{2}\mathbf{i}+y^{2}\mathbf{j}+z^{2}\mathbf{k}\); \(S\) is the surface of the solid enclosed by \(x=0\), \(x=1\), \(y=0\), \(y=1\), \(z=0\) , and \(z=1\).

Question

\(\mathbf{F}=(x-y)\mathbf{i}+(y-z)\mathbf{j}+(x-y)\mathbf{k}\); \( S\) is the surface of the solid cube with its center at the origin and faces in the planes \(x=\pm 1\), \(y=\pm 1\), and \(z=\pm 1\).

Question

\(\mathbf{F}=x^{2}\mathbf{i}+2y\mathbf{j}+4z^{2} \mathbf{k}\); \(S\) is the surface of the solid cylinder \(x^{2}+y^{2}\leq 4\), \(0\leq z\leq 2\).

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\(\mathbf{F}=x\mathbf{i}+2y^{2}\mathbf{j}+3z^{2}\mathbf{k}\); \(S \) is the surface of the solid cylinder \(x^{2}+y^{2}\leq 9\), \(0\leq z\leq 1\).

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\(\mathbf{F}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\); \(S \) is the sphere \(x^{2}+y^{2}+z^{2}=1\).

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\(\mathbf{F}=2x\mathbf{i}+2y\mathbf{j}+2z\mathbf{k}\); \(S\) is the sphere \(x^{2}+y^{2}+z^{2}=2\).

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\(\mathbf{F}=(x+\cos x)\mathbf{i}+(y+y\sin x)\mathbf{j}+2z \mathbf{k}\); \(S\) is the surface of the solid enclosed by the tetrahedron with vertices \((0,0,0),(1,0,0),(0,1,0)\), and \((0,0,1)\).

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\(\mathbf{F}=yz\mathbf{i}+xz\mathbf{j}+xy\mathbf{k}\); \(S\) is the surface of the solid enclosed by the tetrahedron with vertices \((0,0,0),(1,0,0),(0,1,0)\), and \((0,0,1)\).

Question

Volume Let \(\mathbf{F}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\); let \(S\) be the surface of a solid \(E \) satisfying the Divergence Theorem; and let \(\mathbf{n}\) be the outer unit normal vector to \(S\). Show that the volume \(V\) of \(E \) is given by the formula \(V=\dfrac{1}{3}\iint\limits_{S} \mathbf{F}\,{\bf\cdot}\, \mathbf{n}\,dS\).

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A rectangular parallelepiped with sides of length \(a,b\), and \(c\)

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A right circular cone with height \(h\) and base radius \(R\)

(Hint: The calculation is simplified with the cone oriented as shown in the figure.)

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A sphere of radius \(R\)

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\( \mathbf{F}(x,y,z)=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\); \(S\) is the surface of the sphere \(x^{2}+y^{2}+z^{2}=100.\)

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\( \mathbf{F}(x,y,z)=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\); \(S\) is the closed cylindrical surface \(x^{2}+y^{2}=1\) between \(z=0\) and \(z=2.\)

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Show that if \(\mathbf{F}\) is a constant vector field and \(S\) is the surface of a solid \(E \) satisfying the assumptions of the Divergence Theorem, then \(\iint\limits_S \mathbf{F}\,{\bf\cdot}\, \mathbf{n}\,dS=0\).

Question

Let \(\mathbf{F}=3x\mathbf{i}+4y\mathbf{j}+(7z+2x)\mathbf{k}\) and \(\mathbf{G}=2x\mathbf{i}+3y\mathbf{j}+(9z+6y)\mathbf{k}\). Let \(S\) be the surface of a solid \(E \) satisfying the assumptions of the Divergence Theorem. Show that \(\iint\limits_S \mathbf{F}\,{\bf\cdot}\, \mathbf{n} \,dS=\iint\limits_S \mathbf{G}\,{\bf\cdot}\, \mathbf{n}\,dS\).

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Flux Suppose the velocity of a fluid flow in space is constant.

1. Show that the flux through any closed surface is \(0\).
2. In your own words, give a physical interpretation of (a).

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Let \(\mathbf{F}(x,y,z)=\dfrac{1}{\rho }(x\mathbf{i}+y\mathbf{j}+ z\mathbf{k})\), where \(\rho =\sqrt{x^{2}+y^{2}+z^{2}}\). Show that \({\rm{div}} \mathbf{F}=\dfrac{2}{\rho }\).

Question

1. Use Problem 28 and the Divergence Theorem to find \[ \iiint\limits_E \dfrac{dV}{\sqrt{x^{2}+y^{2}+z^{2}}} \] where \(E \) is the solid inside the sphere \(x^{2}+y^{2}+z^{2}=1\).
2. Verify the solution to (a) by finding the triple integral directly.

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Assume that the hypotheses of the Divergence Theorem hold for \(S\) and \(E.\) Show that if \(f\) satisfies the Laplace equation \((f_{xx}+f_{yy}+f_{zz}=0)\) in a closed, bounded solid \(E \) with boundary \(S\) with outer unit normal \(\mathbf{n}\), then \(\iint\limits_S {\bf \nabla} f\,{\bf\cdot}\, \mathbf{n}\,dS=0\).

Question

Let \(\mathbf{F}=x^{2}\mathbf{i}+y^{2}\mathbf{j}+z^{2}\mathbf{k}\) and let \(\mathbf{n}\) be the outer unit normal to the surface \(S\). Use the Divergence Theorem to show that \[ \iint\limits_{S}\mathbf{F}\,{\bf\cdot}\, \mathbf{n}\,dS=\dfrac{8\pi q^{4}}{3} \]

if \(S\) is the surface \(x^{2}+y^{2}+z^{2}=2qz\), \(q>0\).

Question

What is the value of the integral in Problem 31 if \(S\) is the surface of the cube \(x=0\), \(x=q\), \(y=0\), \(y=q\), \(z=0\), and \(z=q\), \(q>0\)?

Question

Let \(f\) and \(g\) be two scalar functions and let \(S\) be the surface of a solid \(E \) satisfying the assumptions of the Divergence Theorem. Show that \[ \iint\limits_{S}f( {\bf\nabla}g) \,{\bf\cdot}\, \mathbf{n\,} dS=\iiint\limits_E[ f( {\bf\nabla}^{2}g) + {\bf\nabla}\! f\,{\bf\cdot}\, {\bf\nabla}g] \,dV \]

[Hint: Let \(\mathbf{F}=f( {\bf\nabla }g) \) in the Divergence Theorem and use \({\bf\nabla}^{2}g={\bf\nabla} \,{\bf\cdot}\, {\bf\nabla} g=g_{xx}+g_{yy}+g_{zz}\).]