Chapter 14

Find the flux of \(\mathbf{F}\) over \(\partial Q\). \(Q\) is bounded by \(z=\sqrt{x^{2}+y^{2}}, x^{2}+y^{2}=1\) and \(z=0\) \(\mathbf{F}=\left\langle y^{2}, x^{2} z, z^{2}\right\rangle\)

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}$$

Assume that \(\iint_{S} \mathbf{D} \cdot \mathbf{n} d S=q,\) for any closed surface \(S,\) where \(\mathbf{D}=\epsilon_{0} \mathbf{E}\) is the electric flux density and \(q\) is the charge enclosed by \(S\). Show that \(\nabla \cdot \mathbf{D}=Q,\) where \(Q\) is the charge density satisfying \(q=\iiint_{R} Q d V\).

Find the surface area of the given surface. The portion of the cone \(z=\sqrt{x^{2}+y^{2}}\) below the plane \(z=4\)

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}.\) \(\mathbf{F}(x, y)=\left\langle x^{2}+1, y^{3}-3 y+2\right\rangle, C\) is the top half-circle from \((-4,0)\) to \((4,0)\)

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(\mathbf{F}=\left\langle x^{3}-y^{4}, e^{x^{2}+z^{2}}, x^{2}-16 y^{2} z^{2}\right\rangle\) and \(C\) is \(x^{2}+z^{2}=1\) in the plane \(y=0\)

\(C\) is the intersection of \(z=x^{2}+y^{2}\) and \(z=8-y,\) oriented clockwise from above, \(\mathbf{F}=\left\langle 2 x^{2}, 4 y^{2}, e^{8 z^{2}}\right\rangle\) \(C\) is the intersection of \(x^{2}+y^{2}=1\) and \(z=x-y,\) oriented clockwise from above, \(\mathbf{F}=\left\langle\cos x^{2}, \sin y^{2}, \tan z^{2}\right\rangle\)

Determine whether the given vector field is conservative and/or incompressible. $$\left\langle 2 x z, 3 y, x^{2}-y\right\rangle$$

Find the flux of \(\mathbf{F}\) over \(\partial Q\). \(Q\) is bounded by \(z=x^{2}+y^{2}\) and \(z=8-x^{2}-y^{2}\) \(\mathbf{F}=\left\langle 3 y^{2}, 4 x^{3}, 2 z-x^{2}\right\rangle\)

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y, z)=x y z$$

The moment of inertia about the \(z\) -axis of a solid \(Q\) with constant density \(\rho\) is \(I_{z}=\iiint\left(x^{2}+y^{2}\right) \rho d V .\) Express this as a surface integral.

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}.\) \(\mathbf{F}(x, y)=\left\langle x e^{x^{2}}-2, \sin y\right\rangle, C\) is the portion of the parabola \(y=x^{2}\) from \((-2,4)\) to \((2,4)\)

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(\mathbf{F}=\left\langle x^{3}-y^{2} z, \sqrt{x^{2}+z^{2}}, 4 x y-z^{4}\right\rangle\) and \(C\) is formed by \(z=1-x^{2}\) and \(z=0\) in the plane \(y=2\)

Evaluate the line integral. \(\int_{C} 2 y d s,\) where \(C\) is the portion of \(y=x^{2}\) from (0,0) to (2, 4), followed by the line segment to (3,0)

Use Stokes' Theorem to evaluate \(\int c \mathbf{F} \cdot d \mathbf{r}\). \(C\) is the triangle from (0,1,0) to (0,0,4) to (2,0,0) \(\mathbf{F}=\left\langle x^{2}+2 x y^{3} z, 3 x^{2} y^{2} z-y, x^{2} y^{3}\right\rangle\)

Determine whether the given vector field is conservative and/or incompressible. $$\left(x y^{2}, 3 x z, 4-z y^{2}\right)$$

Find the flux of \(\mathbf{F}\) over \(\partial Q\). \(Q \quad\) is bounded by \(\quad x^{2}+z^{2}=1, y=0 \quad\) and \(\quad y=1\) \(\mathbf{F}=\left\langle z-y^{3}, 2 y-\sin z, x^{2}-z\right\rangle\)

Let \(u\) be a scalar function with continuous second partial derivatives. Define the normal derivative \(\frac{\partial u}{\partial n}=\nabla u \cdot \mathbf{n} .\) Show that \(\iint_{S} \frac{\partial u}{\partial n} d S=\iiint_{Q} \nabla^{2} u d V\).

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y, z)=x^{2} y+y z$$

Use a line integral to compute the area of the given region. The ellipse \(4 x^{2}+y^{2}=16\)

Evaluate the line integral. \(\int_{C} 4 z d s,\) where \(C\) is the line segment from (1,0,1) to (2,-2,2)

Use Stokes' Theorem to evaluate \(\int c \mathbf{F} \cdot d \mathbf{r}\). \(C\) is the square from (0,2,2) to (2,2,2) to (2,2,0) to (0,2,0) \(\mathbf{F}=\left\langle x^{2}, y^{3}+x, 3 y^{2} \cos z\right\rangle\)

Determine whether the given vector field is conservative and/or incompressible. $$\langle x, y, 1-3 z\rangle$$

Find the flux of \(\mathbf{F}\) over \(\partial Q\). \(Q\) is bounded by \(y^{2}+z^{2}=4, x=1\) and \(x=8-y\) \(\mathbf{F}=\left\langle x^{2} z, 2 y-e^{z}, \sin x\right\rangle\)

Suppose that \(u\) is a harmonic function (that is, \(\nabla^{2} u=0\) ). Show that \(\iint_{S} \frac{\partial u}{\partial n} d S=0\).

Find the gradient field corresponding to \(f\) Use a CAS to graph it. $$f(x, y, z)=(x-y)^{2}+z$$

Use a line integral to compute the area of the given region. The ellipse \(4 x^{2}+y^{2}=4\)

Evaluate the line integral. \(\int_{C} x z d s,\) where \(C\) is the line segment from (2,1,0) to (2,0,2)

Use Stokes' Theorem to evaluate \(\int c \mathbf{F} \cdot d \mathbf{r}\). \(C\) is the intersection of \(z=4-x^{2}-y^{2}\) and \(x^{2}+z^{2}=1\) with \(y>0,\) oriented clockwise as viewed from the right, \(\mathbf{F}=\left\langle x^{2}+3 y, \cos y^{2}, z^{3}\right\rangle\)

Determine whether the given vector field is conservative and/or incompressible. $$\left\langle 4 x, 3 y^{3}, e^{z}\right\rangle$$

Find the flux of \(\mathbf{F}\) over \(\partial Q\). \(Q\) is bounded by \(y^{2}+z^{2}=4, x=1\) and \(x=8-y\) \(\mathbf{F}=\left\langle x^{2} z, 2 y-e^{z}, \sin x\right\rangle\)

Determine whether or not the vector field is conservative. If it is, find a potential function. $$\langle y, x\rangle$$

If the heat conductivity \(k\) is not constant, our derivation of the heat equation is no longer valid. If \(k=K(x, y, z),\) show that the heat equation becomes \(K \nabla^{2} T+\nabla K \cdot \nabla T=\sigma \rho \frac{\partial T}{\partial t}\).

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}.\) \(\mathbf{F}(x, y, z)=\frac{\langle x, y, z\rangle}{\sqrt{x^{2}+y^{2}+z^{2}}}, C\) runs from \((1,3,2)\) to \((2,1,5)\)

Use a line integral to compute the area of the given region. The region bounded by \(x^{2 / 3}+y^{2 / 3}=1 .\) (Hint: Let \(x=\cos ^{3} t\) and \(\left.y=\sin ^{3} t\right)\)

Evaluate the line integral. \(\int_{C} 4(x-z) z d x,\) where \(C\) is the portion of \(y=x^{2}\) in the plane \(z=2\) from (1,1,2) to (2,4,2)

Use Stokes' Theorem to evaluate \(\int c \mathbf{F} \cdot d \mathbf{r}\). \(C\) is the intersection of \(z=x^{2}+y^{2}-4\) and \(z=y-1\) oriented clockwise as viewed from above, \(\mathbf{F}=\left\langle\sin x^{2}, y^{3}, z \ln z-x\right\rangle\)

Determine whether the given vector field is conservative and/or incompressible. $$\left(\sin x, 2 y^{2}, \sqrt{2}\right)$$

Find the flux of \(\mathbf{F}\) over \(\partial Q\). \(Q\) is bounded by \(y=4-x^{2}-z^{2}\) and the \(x z\) -plane, \(\mathbf{F}=\left\langle z^{2} x, x^{2} y, y^{2} x\right\rangle\)

If \(h\) has continuous partial derivatives and \(S\) is a closed surface enclosing a solid \(Q,\) show that \(\iint_{S}(h \nabla h) \cdot \mathbf{n} d S=\iiint_{Q}\left(h \nabla^{2} h+\nabla h \cdot \nabla h\right) d V\).

Determine whether or not the vector field is conservative. If it is, find a potential function. $$\langle 2, y\rangle$$

Find the surface area of the given surface. The portion of the paraboloid \(z=x^{2}+y^{2}\) inside the cylinder \(x^{2}+y^{2}=4\)

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}.\) \(\mathbf{F}(x, y, z)=\frac{\langle x, y, z\rangle}{x^{2}+y^{2}+z^{2}}, C\) runs from \((2,0,0)\) to \((0,1,-1)\)

Use a line integral to compute the area of the given region. The region bounded by \(x^{2 / 5}+y^{2 / 5}=1\)

Evaluate the line integral. \(\int_{C} z d s,\) where \(C\) is the intersection of \(x^{2}+y^{2}=4\) and \(z=0\) (oriented clockwise as viewed from above)

Determine whether the given vector field is conservative and/or incompressible. $$\left(-2 x y, z^{2} \cos y z^{2}-x^{2}, 2 y z \cos y z^{2}\right)$$

Find the flux of \(\mathbf{F}\) over \(\partial Q\). \(Q\) is bounded by \(3 x+2 y+z=6\) and the coordinate planes, \(\mathbf{F}=\left\langle y^{2} x, 4 x^{2} \sin z, 3\right\rangle\)

Determine whether or not the vector field is conservative. If it is, find a potential function. $$\langle y,-x\rangle$$

Suppose that \(f\) and \(g\) are both harmonic (that is, \(\left.\nabla^{2} f=\nabla^{2} g=0\right)\) and \(f=g\) on a closed surface \(S,\) where \(S\) en closes a solid \(Q .\) Use the result of exercise \(24,\) with \(h=f-g\) to show that \(f=g\) in \(Q\).

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}.\) \(\mathbf{F}(x, y)=\left\langle 3 x^{2} y+1,3 x y^{2}\right\rangle, C\) is the bottom half-circle from \((1,0)\) to \((-1,0)\)