Answer: Heat transfer Fourier’s Law of heat transfer (or

Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. \(\mathbf{F}=\left\langle x^{2}, y^{2}, z^{2}\right)\); S is the sphere \(\left\{(x, y, z): x^{2}+y^{2}+z^{2}=25\right\}\) Text Transcription: F = langle x^2, y^2, z^2 rangle {(x, y, z): x^2 + y^2 + z^2 = 25}

Gauss' Law for gravitation The gravitational force due to a point mass M is proportional to \(\mathbf{F}=G M \mathbf{r} /|\mathbf{r}|^{3}\), where \(\mathbf{r}=\langle x, y, z)\) and G is the gravitational constant. a. Show that the flux of the force field across a sphere of radius a centered at the origin is \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi G M\). b. Let S be the boundary of the region between two spheres centered at the origin of radius a and b with a < b. Use the Divergence Theorem to show that the net outward flux across S is zero. c. Suppose there is a distribution of mass within a region D containing the origin. Let p(.r. _v. _) be the mass density (mass per unit volume). Interpret the sta tement that \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi G \iiint_{D} \rho(x, y, z) d V\) d. Assuming F satisfies the conditions of the Divergence Theorem, conclude from pan (c) that \(\nabla \cdot \mathbf{F}=4 \pi G \rho\). e. Because the gravitational force is conservative, it has a potential function \(\varphi\). From pan (d) conclude that \(\nabla^{2} \varphi=4 \pi G \rho\). Text Transcription: F = GMr|r|^3 r = langle x, y, z rangle iint_S F cdot n dS = 4piGM iint_S f cdot n dS = 4piG iint_D p(x,y, z) dV Nabla cdot F = 4piGp Varphi nabla^2 varhpi = 4piGp

Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. \(\mathbf{F}=\langle x,-2 y, 3 z\rangle\); S is the sphere \(\left\{(x, y, z): x^{2}+y^{2}+z^{2}=6\right\}\) Text Transcription: F = langle x, -2y, 3z rangle {(x, y, z): x^2 + y^2 + z^2 = 6}

Rotation fields Find the net outward flux of F = a X r across any smooth closed surface in \(\mathbf{R}^{3}\). where a is a constant nonzero vector and\(\mathbf{r}=\langle x, y, z\rangle\). Text Transcription: R^3 r = langle x, y, z rangle

Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. \(\mathbf{F}=\left\langle y-2 x, x^{3}-y, y^{2}-z\right\rangle\); S is the sphere \(\left\{(x, y, z): x^{2}+y^{2}+z^{2}=4\right\}\) Text Transcription: F = langle y - 2x, x^3 - y, y^2 - z rangle {(x, y, z): x^2 + y^2 + z^2 = 4}

Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. \(\mathbf{F}=\left\langle x^{2}, 2 x z, y^{2}\right\rangle\); S is the surface of the cube cut from the first octant by the planes x = 1, y = 1, and z = 1. Text Transcription: rangle F = langle x^2, 2xz, y^2

Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. \(\mathbf{F}=\langle x, 2 y, z\rangle\); S is the boundary of the tetrahedron in the first octant formed by the plane x + y + z = 1. Text Transcription: F = langle x, 2y, z rangle

Heal transfer Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector F al a point is proportional To the negative gradient of the temperature: that is, \(\mathbf{F}=-k \nabla T\). which means that heal energy flows fmnt hot regions to cold regions. The constant k is called the conductivity which has metric units of J/m-s-K or W/m-K. A temperature function for a region D is given. Find the net outward heat flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=-k \iint_{S} \nabla T \cdot \mathbf{n} d S\) across the boundary S of D. In some cases it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. \(T(x, y, z)=100+x+2 y+z\); \(D=\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\}\) Text Transcription: F = -k nabla T iint_S F cdot n dS = -k iint_S nablaT cdot n dS T(x, y, z) = 100 + x + 2y + z D = {(x, y, z): 0 leq x leq 1, 0 leq y leq 1, 0 leq z leq 1}

Heal transfer Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector F al a point is proportional To the negative gradient of the temperature: that is, \(\mathbf{F}=-k \nabla T\). which means that heal energy flows fmnt hot regions to cold regions. The constant k is called the conductivity which has metric units of J/m-s-K or W/m-K. A temperature function for a region D is given. Find the net outward heat flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=-k \iint_{S} \nabla T \cdot \mathbf{n} d S\) across the boundary S of D. In some cases it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. \(T(x, y, z)=100+x^{2}+y^{2}+z^{2}\); \(D=\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\}\) Text Transcription: F = -k nabla T iint_S F cdot n dS = -k iint_S nablaT cdot n dS T(x, y, z) = 100 + x^2 + y^2 + z^2 D = {(x, y, z): 0 leq x leq 1,0 leq z leq 1}

Explain the meaning of Lhe surface integral in the Divergence Theorem.

Explain the meaning of the volume integral in the Divergence Theorem.

Explain the meaning of the Divergence Theorem.

What is the net outward flux of the rotation field \(\mathbf{F}=\langle 2 z+y,-x,-2 x\rangle\) across the surface that encloses any region? Text Transcription: F = langle 2z + y, -x, -2x rangle

What is the net outward flux of the radial field \(\mathbf{F}=\langle x, y, z\rangle\) across the sphere of radius 2 centered at the origin? Text Transcription: F = langle x, y, z rangle

What is the divergence of an inverse square vector field?

Suppose div F = 0 in a region enclosed by two concentric spheres. What is the relationship between the outward fluxes across the two spheres?

If div F > 0 in a region enclosed by a small cube, is the net flux of the field into or out of the cube?

\(\mathbf{F}=\langle 2 x, 3 y, 4 z\rangle\); \(D=\left\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 4\right\}\) Text Transcription: F = langle 2x, 3y, 4z rangle D = {(x, y, z): x^2 + y^2 + z^2 leq 4}

\(\mathbf{F}=\langle-x,-y,-z\rangle\); \(D=\{(x, y, z):|x| \leq 1,|y| \leq 1,|z| \leq 1\}\) Text Transcription: F = langle -x, -y, -z rangle D = {(x, y, z): |x| leq 1, |y| leq 1, |z| leq 1}

\(\mathbf{F}=\langle z-y, x,-x\rangle\); \(D=\left\{(x, y, z): x^{2} / 4+y^{2} / 8+z^{2} / 12 \leq 1\right\}\) Text Transcription: F = langle z - y, x, - x rangle D = {(x, y, z): x^2/4 + y^2/8 + z^2/12 leq 1}

\(\mathbf{F}=\left\langle x^{2}, y^{2}, z^{2}\right\rangle\); \(D=\{(x, y, z):|x| \leq 1,|y| \leq 2,|z| \leq 3\}\) Text Transcription: F = langle x^2, y^2, z^2 rangle D = {(x, y, z): |x| leq 1, |y| leq 2, |z| leq 3}

Rotation fields Find the net outward llux of the field \(\mathbf{F}=\langle 2 z-y, x,-2 x\rangle\) across the sphere of radius I centered at the origin. Text Transcription: F = langle 2z - y, x, -2x rangle

Rotation fields Find the net outward flux of the field \(\mathbf{F}=\langle z-y, x-z, y-x\rangle\) across the boundary of the cube\(\{(x, y, z):|x| \leq 1,|y| \leq 1,|z| \leq 1\}\). Text Transcription: F = langle z - y, x - z, y - x rangle {(x, y, z): |x| leq 1, |y| leq 1, |z| leq 1 }

Rotation fields Find the net outward flux of the field \(\mathbf{F}=\langle b z-c y, c x-a z, a y-b x\rangle\) across any smooth closed surface in \(\mathbf{R}^{3}\). where a. b. and c are constants. Text Transcription: F = langle bz - cy, cx - az, ay - bx rangle R^3

Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. \(\mathbf{F}=\langle y+z, x+z, x+y\rangle\); S consists of the faces of the cube \(\{(x, y, z):|x| \leq 1,|y| \leq 1,|z| \leq 1\}\) Text Transcription: F = langle y + z, x + z, x + y rangle {(x, y, z): |x| leq 1, |y| leq 1, |z| leq 1}

Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. \(\mathbf{F}=\langle x, y, z\rangle\); S is the surface of the paraboloid \(z=4-x^{2}-y^{2}\), for \(z \geq 0\), plus its base in the xy-plane. Text Transcription: F = langle x, y, z rangle z = 4 - x^2 - y^2 Z leq 0

Computing flux Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces S. \(\mathbf{F}=\langle x, y, z\rangle\); S is the surface of the cone \(z^{2}=x^{2}+y^{2}\), for \(0 \leq z \leq 4\), plus its top surface in the plane z = 4. Text Transcription: F = langle x, y, z rangle z^2 = x^2 + y^2 0 leq z leq 4

Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. \(\mathbf{F}=\langle z-x, x-y, 2 y-z\rangle\); D is the region between the spheres of radius 2 and 4 centered at the origin, Text Transcription: F = langle z - x, x - y, 2y - z rangle

Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. \(\mathbf{F}=\mathbf{r}|\mathbf{r}|=\langle x, y, z\rangle \sqrt{x^{2}+y^{2}+z^{2}}\); D is the region between the spheres of radius 1 and 2 centered at the origin. Text Transcription: F = r|r| = langle x, y, z rangle sqrt x^2 + y^2 + z^2

Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|}=\frac{\langle x, y, z\rangle}{\sqrt{x^{2}+y^{2}+z^{2}}}\); D is the region between the spheres of radius 1 and 2 centered at the origin. Text Transcription: F = r/|r| = langle x, y, z rangle/ sqrt x^2 + y^2 + z^2

Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. \(\mathbf{F}=\langle z-y, x-z, 2 y-x\rangle\); d is the region between two cubes: \(\{(x, y, z): 1 \leq|x| \leq 3,1 \leq|y| \leq 3,1 \leq|z| \leq 3\}\) Text Transcription: F = langle z - y, x - z, 2y - x rangle {(x, y, z): 1 leq |x| leq 3, 1 leq |y| leq 3, 1 leq |z| leq 3}

Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. \(\mathbf{F}=\left\langle x^{2},-y^{2}, z^{2}\right\rangle\); D is the region in the first octant between the plance z = 4 - x - y and z = 2 - x - y. Text Transcription: F = langle x^2, -y^2, z^2 rangle

Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D. \(\mathbf{F}=\langle x, 2 y, 3 z\rangle\); D is the region between the cylinders \(x^{2}+y^{2}=1\) and \(x^{2}+y^{2}=4\), for \(0 \leq z \leq 8\). Text Transcription: F = langle x, 2y, 3z rangle x^2 + y^2 = 1 x^2 + y^2 = 4 0 leq z leq 8

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If \(\nabla \cdot \mathbf{F}=0\) at all points of a region D. then \(\mathbf{F} \cdot \mathbf{n}=0\) at all points of the boundary of D. b. If .\(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=0\) on all closed surfaces in \(\mathbf{R}^{3}\), then F is constant. c. If |F| < l. then \(\left|\iiint_{D} \nabla \cdot \mathbf{F} d V\right|\) dvl is less than the area of the surface of D. Text Transcription: Nabla cdot F = 0 F cdot = 0 iint_SF cdot ndS = 0 |iiint_D nabla cdot F dV|

Flux across a sphere Consider the radial field \(\mathbf{F}=\langle x, y, z\rangle\) and let S be the sphere of radius a centered at the origin. Compute the outward flux of F across S using the representation \(z=\pm \sqrt{a^{2}-x^{2}-y^{2}}\) for the sphere (either symmetry or two surfaces must be used). Text Transcription: F = langle x, y, z rangle Z = pm sqrt a^2 - x^2 - y^2

Flux integrals Compute the oufwardflux of the following vector fields across the given surfaces S. You should decide which integral Qf the Divergence Theorem Io use. \(\mathbf{F}=\left\langle x^{2} e^{y} \cos z,-4 x e^{y} \cos z, 2 x e^{y} \sin z\right\rangle\); S is the boundary of the ellipsoid \(x^{2} / 4+y^{2}+z^{2}=1\) Text Transcription: F = langle x^2e^y cos z, -4xe^y cos z, 2xe^y sin z rangle\ x^2/4 + y^2 + z^2 = 1

Flux integrals Compute the oufwardflux of the following vector fields across the given surfaces S. You should decide which integral Qf the Divergence Theorem Io use. \(\mathbf{F}=\langle-y z, x z, 1\rangle\); S is the boundary of the ellipsoid \(x^{2} / 4+y^{2} / 4+z^{2}=1\). Text Transcription: F = langle -yz, xz, 1 rangle x^2/4 + y^2/4 + z^2 = 1

Flux integrals Compute the oufwardflux of the following vector fields across the given surfaces S. You should decide which integral Qf the Divergence Theorem Io use. \(\mathbf{F}=\langle x \sin y,-\cos y, z \sin y\rangle\); S is the boundary of the region bounded by the planes x = 1, y = 0, \(y=\pi / 2\), z = 0, and z = x Text Transcription: F = langle x sin y, -cos y, z sin y rangle y = pi/2

Radial fields Consider the radial vector field \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{p / 2}}\), let S be the sphere of radius a centered at the origin. a. Use a surface integral to show that the outward flux of F across S is \(4 \pi a^{3-p}\). Recall that the unit normal lo the sphere is r/ |r|. b. For what values of p does F satisfy the conditions of the Divergence Theorem? For these values of p. use the fact \(\nabla \cdot \mathbf{F}=\frac{3-p}{|\mathbf{r}|^{p}}\) across S using the Divergence Theorem. Text Transcription: F = r / |r|^p = langle x, y, z rangle / (x^2 + y^2 + z^2)^p/2 4pia^3-p Nabla cdot F = 3 - p / |r|^p

Singular radial field Consider the radial field \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}}\) a. Evaluate a surface integral to show that \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi a^{2}\). where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of t are undefined at the origin. so the Divergence Theorem does not apply directly. Nevertheless Lhe flux across the sphere as computed in pan (a) is finite. Evaluate the triple integral of the Divergence Theorem as an improper integral 35 follows. Integrate div F over the region between two spheres of radius a and \(0<\varepsilon<a\). Then let \(\varepsilon \rightarrow 0^{+}\) to obtain the flux computed in pan (a). Text Transcription: F = r / |r| = langle x, y, z rangle / (x^2 + y^2 + z^2)^½ iint_S F cdot ndS =4pia^2 0 < varepsilon < a varepsilon rightarrow 0^+

Logarithmic potential Consider the potential function \(\varphi(x, y, z)=\frac{1}{2} \ln \left(x^{2}+y^{2}+z^{2}\right)=\ln |\mathbf{r}|\), where \(\mathbf{r}=\langle x, y, z\rangle\) a Show that the gradient field associated with \(\varphi\) is \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{2}}=\frac{\langle x, y, z\rangle}{x^{2}+y^{2}+z^{2}}\) b. Show that\(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi a\), where S is the surface of a sphere of radius a centered at the origin. c. Compute div F. d. Note that F is undefined at the origin. so the Divergence Theorem does not apply directly. Evaluate the volume integral as described in Exercise 37. Text Transcription: varphi(x, y, z)= ½ ln (x^2 + y^2 + z^2) = ln |r| r = langle x, y, z rangle iint_S F cdot n dS = 4pia varphi

Gauss' Law for electric fields The electric field due no a point charge Q is \(\mathbf{E}=\frac{Q}{4 \pi \varepsilon_{0}} \frac{\mathbf{r}}{|\mathbf{r}|^{3}}\), where \(\mathbf{r}=\langle x, y, z\rangle\), and \(\varepsilon_{0}\) is a constant. a. Show that the flux of the field across a sphere of radius a centered at the origin is \(\iint_{S} \mathbf{E} \cdot \mathbf{n} d S=\frac{Q}{\varepsilon_{0}}\). b. Let S be the boundary of the region between two spheres centered at the origin of radius a and b with a < b. Use the Divergence Theorem to show that the net outward flux across S is zero. c. Suppose there is a distribution of charge within a region D. Let q(x, y, z) be the charge density (charge per unit volume). Interpret the statement that \(\iint_{S} \mathbf{E} \cdot \mathbf{n} d S=\frac{1}{\varepsilon_{0}} \iiint_{D} q(x, y, z) d V\) d. Assuming E satisfies the conditions of the Divergence q Theorem. conclude from pan (c) that \(\nabla \cdot \mathbf{E}=\frac{q}{\varepsilon_{0}}\) e. Because the electric force is conservative, it has a potential . 7 q function \(\varphi\). From pan (d) conclude that \(\nabla^{2} \varphi=\nabla \cdot \nabla \varphi=\frac{q}{\varepsilon_{0}}\) Text Transcription: iint_S E cdot n dS = 1/varepsilon_0 iiint_D q(x, y, z)dV nabla cdot E = q/ varepsilon_0 nabla^2 varhpi = nabla cdot nabla_varphi = 9/ varepsilon_0 iint_S E cdot n dS = Q/varepsilon_0 E = Q/4pi varepsilon_0 r/|r|^3 r = langle x, y, z rangle

Heal transfer Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector F al a point is proportional To the negative gradient of the temperature: that is, \(\mathbf{F}=-k \nabla T\). which means that heal energy flows fmnt hot regions to cold regions. The constant k is called the conductivity which has metric units of J/m-s-K or W/m-K. A temperature function for a region D is given. Find the net outward heat flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=-k \iint_{S} \nabla T \cdot \mathbf{n} d S\) across the boundary S of D. In some cases it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. \(T(x, y, z)=100+e^{-z}\); \(D=\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\}\) Text Transcription: F = -k nabla T iint_S F cdot n dS = -k iint_S nablaT cdot n dS T(x, y, z) = 100 + e^-1 D = {(x, y, z): 0 leq x leq 1, 0 leq y leq 1, 0 leq z leq 1}

Heal transfer Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector F al a point is proportional To the negative gradient of the temperature: that is, \(\mathbf{F}=-k \nabla T\). which means that heal energy flows fmnt hot regions to cold regions. The constant k is called the conductivity which has metric units of J/m-s-K or W/m-K. A temperature function for a region D is given. Find the net outward heat flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=-k \iint_{S} \nabla T \cdot \mathbf{n} d S\) across the boundary S of D. In some cases it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. \(T(x, y, z)=100+x^{2}+y^{2}+z^{2}\); D is the unit sphere centered at the origin. Text Transcription: F = -k nabla T iint_S F cdot n dS = -k iint_S nablaT cdot n dS T(x, y, z) = 100 + x^2 +y^2 + z^2

Heal transfer Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector F al a point is proportional To the negative gradient of the temperature: that is, \(\mathbf{F}=-k \nabla T\). which means that heal energy flows fmnt hot regions to cold regions. The constant k is called the conductivity which has metric units of J/m-s-K or W/m-K. A temperature function for a region D is given. Find the net outward heat flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=-k \iint_{S} \nabla T \cdot \mathbf{n} d S\) across the boundary S of D. In some cases it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. \(T(x, y, z)=100 e^{-x^{2}-y^{2}-z^{2}}\); D is the sphere of radius a centered at the origin. Text Transcription: F = -k nabla T iint_S F cdot n dS = -k iint_S nablaT cdot n dS T(x, y, z) = 100e^-x^2 -y^2 -z^2

Inverse square fields are special Let F be a radial Held \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\). where p is a real number and \(\mathbf{r}=\langle x, y, z\rangle\). With p = 3, F is an inverse square field. a. Show that the net flux across a sphere centered at the origin is independent of the radius of the sphere only for p = 3. b. Explain the observation in pan (a) by finding the flux of \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{P}\) across the boundaries of a spherical box \(\left\{(\rho, \varphi, \theta): a \leq \rho \leq b, \varphi_{0} \leq \varphi \leq \varphi_{1}, \theta_{1} \leq \theta \leq \theta_{2}\right\}\) for various values of p. Text Transcription: F = r/|r|^p r = langle x, y, z rangle {(rho, varphi, theta): a leq rho leq b, varhpi_0 leq varphi leq varphi_1, theta_1 leq theta leq theta_2}

beautiful flux integral Consider the potential function \(\varphi(x, y, z,)=G(\rho)\), where G is any twice differentiable function and \(\rho=\sqrt{x^{2}+y^{2}+z^{2}}\) therefore. G depends only on the distance from the origin. a. Show that the gradient vector field associated with \(\varphi\) is \(\mathbf{F}=\nabla \varphi=G^{\prime}(\rho) \frac{\mathbf{r}}{\rho}\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(\rho=|\mathbf{r}|\). b. Let S be the sphere of radius a centered at the origin and let D be.the region enclosed by S. Show that the flux of F across S is \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d A=4 \pi a^{2} G^{\prime}(a)\) c. Show that \(\nabla \cdot \mathbf{F}=\nabla \cdot \nabla \varphi=\frac{2 G^{\prime}(\rho)}{\rho}+G^{\prime \prime}(\rho)\) d. Use pan (c) to show that the flux across S (as given in pan (b) ) is also obtained by the volume integral \(\iiint_{D} \nabla \cdot \mathbf{F} d V\). (Hint: use spherical coordinates and integrate by parts.) Text Transcription: varphi(x, y, z,) = G(p) rho = sqrt x^2 + y^2 + z^2 F = nabla_varphi = G’(rho)r/rho r = langle x, y, z rangle rho = |r| Nabla cdot F = nabla cdot nabla_varphi = 2G’(rho)/rho + G”(rho) iint_D nabla cdot F dV

Integration by parts (Gauss' formula) Recall the Product Rule or Theorem 14.1 l:\(\nabla \cdot(u \mathbf{F})=u \nabla \cdot \mathbf{F}+\mathbf{F} \cdot \nabla u\).. a. Integrate both sides of this identity over a solid region D with a closed boundary S and use the Divergence Theorem to prove an integration by parts rule: \(\iiint_{D} u \nabla \cdot \mathbf{F} d V=\iint_{S} u \mathbf{F} \cdot \mathbf{n} d S-\iiint_{D} \mathbf{F} \cdot \nabla u d V\) b. Explain the correspondence between this rule and the integration by parts rule for single-variable functions. c. Use integration by parts to evaluate \(\iiint_{D}\left(x^{2} y+y^{2} z+z^{2} x\right) d V\), where D is the cube in the first octant cut by the planes x = 1, y = 1, and z = 1.. Text Transcription: Nabla cdot (uF) = u nabla cdot F + F cdot nabla u iiint_D u nabla cdot F dV = iint_S uF cdot n dS - iiint_D F cdot nabla u dV iiint_D(x^2y+ y^2z + z^2x) dV

Green's First Identity Prove Green's First Identity for twice differentiable scalar-valued functions u and v defined on a region D: \(\iiint_{D}\left(u \nabla^{2} v+\nabla u \cdot \nabla v\right) d V=\iint_{S} u \nabla v \cdot \mathbf{n} d S\) where \(\nabla^{2} v=\nabla \cdot \nabla v\). You may apply Gauss' Formula in Exercise 48 to \(\mathbf{F}=\nabla v\) or apply the Divergence Theorem to \(\mathbf{F}=u \nabla v\). Text Transcription: nabla^2v = nabla cdot nabla v F = nabla v F = u nabla v iiint_D(u nabla^2 v + nabla u cdot nabla v) dV = iint_S u nabla v cdot n dS

Green's Second Identity Prove Green's Second Identity for scalar-valued functions u and v defined on a region D: \(\iiint_{D}\left(u \nabla^{2} v-v \nabla^{2} u\right) d V=\iint_{S}(u \nabla v-v \nabla u) \cdot \mathbf{n} d S\) (Hint: Reverse the roles of II and v in Green's First Identity.) Text Transcription: iiint_D (u nabla^2 v - v nabla^2 u)dV = iint_S (u nabla v - v nabla u) cdot n dS

Harmonic functions A scalar-valued function \(\varphi\) is harmonic on a region D if \(\nabla^{2} \varphi=\nabla \cdot \nabla \varphi=0\) at ull points of D. Show that the potential function \(\varphi(x, y, z)=|\mathbf{r}|^{-p}\) is harmonic provided p = 0 or p = 1, where \(\mathbf{r}=\langle x, y, z\rangle\). To what vector fields do these potentials correspond? Text Transcription: nabla^2varphi = nabla cdot nabla_varphi = 0 varphi(x, y, z) = |r|^-p r = langle x, y, z rangle

Harmonic functions A scalar-valued function \(\varphi\) is harmonic on a region D if \(\nabla^{2} \varphi=\nabla \cdot \nabla \varphi=0\) at ull points of D. Show that if 4) is harmonic on a region D enclosed by a surface S. then \(\iint_{S} \nabla \varphi \cdot \mathbf{n} d S=0\). Text Transcription: nabla^2varphi = nabla cdot nabla_varphi = 0 iint_S nabla_varphi cdot n dS = 0

Harmonic functions A scalar-valued function \(\varphi\) is harmonic on a region D if \(\nabla^{2} \varphi=\nabla \cdot \nabla \varphi=0\) at ull points of D. Show that if u is harmonic on a region D enclosed by a surface S. then \(\iint_{S} u \nabla u \cdot \mathbf{n} d S=\iiint_{D}|\nabla u|^{2} d V\). Text Transcription: nabla^2varphi = nabla cdot nabla_varphi = 0 iint_S u nabla u cdo n dS = iiint_D |nabla u|^2 dV

Miscellaneous integral identities Prove the following identities. a. \(\iiint_{D} \nabla \times \mathbf{F} d V=\iint_{S}(\mathbf{n} \times \mathbf{F}) d S\) (Hint: Apply me Divergence Theorem to each component of the identity.) b. \(\iint_{S}(\mathbf{n} \times \nabla \varphi) d S=\oint_{C} \varphi d \mathbf{r}\)Hin1: Apply Stokes' Theorem lo each component of the identity.) Text Transcription: iiint_D nabla x F dV = iint_S (n x F) dS iint_S (n x nabla_varphi) dS = oint_c varphi dr

Green's Formula Write Gauss' Formula of Exercise 48 in two dimensions-that is, where \(\mathbf{F}=\langle f, g\rangle\), D is a plane region R and C is the boundary of R. Show that the result is Green's Formula: \(\iint_{R} u\left(f_{x}+g_{y}\right) d A=\oint_{C} u(\mathbf{F} \cdot \mathbf{n}) d s-\iint_{R}\left(f u_{x}+g u_{y}\right) d A\) Show that with II = I . one form of Green's Theorem appears. Which form of Green's Theorem is it? Text Transcription: F = langle f, g rangle iint_R u(f_x + g_y) dA = oint_C u(F cdot n)ds - iint_R(fu_x + gu_y)dA