Chapter 14

Evaluate each line integral. \(\int_{C} y^{3} d x+x^{3} d y ; C\) is the right-angle curve from \((-4,1)\) to \((-4,-2)\) to \((2,-2)\).

In Problems 7-12, find \(\nabla f\). $$ f(x, y, z)=\frac{1}{2}\left(x^{2}+y^{2}+z^{2}\right) $$

In Problems 1-14, use Gauss's Divergence Theorem to calculate \(\iint_{\partial S} \mathbf{F} \cdot \mathbf{n} d S .\) \(\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k} ; S\) is the solid enclosed by \(x+y+z=4, x=0, y=0, z=0\).

\(\int_{C} y^{3} d x+x^{3} d y ; C\) is the curve \(x=2 t, \quad y=t^{2}-3\), \(-2 \leq t \leq 1 .\)

Evaluate each line integral. \(\int_{C} y^{3} d x+x^{3} d y ; C\) is the curve \(x=2 t, y=t^{2}-3\), \(-2 \leq t \leq 1 .\)

In Problems \(7-12\), use Stokes's Theorem to calculate \(\oint_{C} \mathbf{F} \cdot \mathbf{T} d s\). \(\mathbf{F}=(z-y) \mathbf{i}+y \mathbf{j}+x \mathbf{k} ; C\) is the intersection of the cylinder \(x^{2}+y^{2}=x\) with the sphere \(x^{2}+y^{2}+z^{2}=1\), oriented counterclockwise as viewed from above.

In Problems 7-12, find \(\nabla f\). $$ f(x, y, z)=x e^{y} \cos z $$

In Problems 1-14, use Gauss's Divergence Theorem to calculate \(\iint_{\partial S} \mathbf{F} \cdot \mathbf{n} d S .\) \(\mathbf{F}(x, y, z)=2 x \mathbf{i}+3 y \mathbf{j}+4 z \mathbf{k} ; S\) is the solid spherical shell \(9 \leq x^{2}+y^{2}+z^{2} \leq 25\).

Use the vector forms of Green's Theorem to calculate (a) \(\oint_{C} \mathbf{F} \cdot \mathbf{n} d s\) and (b) \(\oint_{C} \mathbf{F} \cdot \mathbf{T} d s\). \(\mathbf{F}=y^{3} \mathbf{i}+x^{3} \mathbf{j} ; C\) is the unit circle.

\(\int_{C}(x+2 y) d x+(x-2 y) d y ; C\) is the line segment from \((1,1)\) to \((3,-1)\).

Evaluate each line integral. \(\int_{C}(x+2 y) d x+(x-2 y) d y ; C\) is the line segment from \((1,1)\) to \((3,-1)\).

In Problems \(7-12\), use Stokes's Theorem to calculate \(\oint_{C} \mathbf{F} \cdot \mathbf{T} d s\). \(\mathbf{F}=(y-z) \mathbf{i}+(z-x) \mathbf{j}+(x-y) \mathbf{k} ; C\) is the ellipse which is the intersection of the plane \(x+z=1\) and the cylinder \(x^{2}+y^{2}=1\), oriented clockwise as viewed from above.

In Problems 7-12, find \(\nabla f\). $$ f(x, y, z)=y^{2} e^{-2 z} $$

In Problems 1-14, use Gauss's Divergence Theorem to calculate \(\iint_{\partial S} \mathbf{F} \cdot \mathbf{n} d S .\) \(\mathbf{F}(x, y, z)=2 z \mathbf{i}+x \mathbf{j}+z^{2} \mathbf{k} ; S\) is the solid cylindrical shell \(1 \leq x^{2}+y^{2} \leq 4,0 \leq z \leq 2\)

Use the vector forms of Green's Theorem to calculate (a) \(\oint_{C} \mathbf{F} \cdot \mathbf{n} d s\) and (b) \(\oint_{C} \mathbf{F} \cdot \mathbf{T} d s\). \(\mathbf{F}=x \mathbf{i}+y \mathbf{j} ; C\) is the unit circle.

$$ \int_{C} y d x+x d y ; C \text { is the curve } y=x^{2}, 0 \leq x \leq 1 $$

Evaluate each line integral. \(\int_{C} y d x+x d y ; C\) is the curve \(y=x^{2}, 0 \leq x \leq 1\).

Suppose that the surface \(S\) is determined by the formula \(z=g(x, y)\). Show that the surface integral in Stokes's Theorem can be written as a double integral in the following way: $$ \iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S=\iint_{S_{x y}}(\operatorname{curl} \mathbf{F}) \cdot\left(-g_{x} \mathbf{i}-g_{y} \mathbf{j}+\mathbf{k}\right) d A $$ where \(\mathbf{n}\) is the upward normal to \(S\) and \(S_{x y}\) is the projection of \(S\) in the \(x y\)-plane.

In Problems 13-18, find div \(\mathbf{F}\) and curl \(\mathbf{F}\). $$ \mathbf{F}(x, y, z)=x^{2} \mathbf{i}-2 x y \mathbf{j}+y z^{2} \mathbf{k} $$

Suppose that the integrals \(\oint \mathbf{F} \cdot \mathbf{T} d s\) taken counterclockwise around the circles \(x^{2}+y^{2}=36\) and \(x^{2}+y^{2}=1\) are 30 and \(-20\), respectively. Calculate \(\iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{k} d A\), where \(S\) is the region between the circles.

\(\int_{C}(x+y+z) d x+x d y-y z d z ; C\) is the line segment from \((1,2,1)\) to \((2,1,0)\).

Evaluate each line integral. \(\int_{C}(x+y+z) d x+x d y-y z d z ; C\) is the line segment from \((1,2,1)\) to \((2,1,0)\).

In Problems 13-18, find div \(\mathbf{F}\) and curl \(\mathbf{F}\). $$ \mathbf{F}(x, y, z)=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k} $$

\(\int_{C} x z d x+(y+z) d y+x d z ; C\) is the curve \(x=e^{t}\), \(y=e^{-t}, z=e^{2 t}, 0 \leq t \leq 1 .\)

Find the mass of the surface \(z=1-\left(x^{2}+y^{2}\right) / 2\) over \(0 \leq x \leq 1,0 \leq y \leq 1\), if \(\delta(x, y, z)=k x y\).

Evaluate each line integral. \(\int_{C} x z d x+(y+z) d y+x d z ; C\) is the curve \(x=e^{t}\), \(y=e^{-t}, z=e^{2 t}, 0 \leq t \leq 1 .\)

In Problems 13-18, find div \(\mathbf{F}\) and curl \(\mathbf{F}\). $$ \mathbf{F}(x, y, z)=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k} $$

Let \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\), and let \(S\) be a solid for which Gauss's Divergence Theorem applies. Show that the volume of \(S\) is given by $$ V(S)=\frac{1}{3} \iint_{\partial S} \mathbf{F} \cdot \mathbf{n} d S $$

\(\int_{C}(x+y+z) d x+(x-2 y+3 z) d y+\) \((2 x+y-z) d z ; C\) is the line-segment path from \((0,0,0)\) to \((2,0,0)\) to \((2,3,0)\) to \((2,3,4)\).

Find the center of mass of the homogeneous triangle with vertices \((a, 0,0),(0, a, 0)\), and \((0,0, a)\).

Evaluate each line integral. \(\int_{C}(x+y+z) d x+(x-2 y+3 z) d y+\) \((2 x+y-z) d z ; C\) is the line-segment path from \((0,0,0)\) to \((2,0,0)\) to \((2,3,0)\) to \((2,3,4)\).

Let \(\mathbf{F}=2 \mathbf{i}+x z \mathbf{j}+z^{3} \mathbf{k}\) and \(\partial S\) be the boundary of the surface \(z=x^{2} y^{2}, x^{2}+y^{2} \leq a^{2}\), oriented counterclockwise as viewed from above. Evaluate \(\oint_{\partial S} \mathbf{F} \cdot \mathbf{T} d s\).

In Problems 13-18, find div \(\mathbf{F}\) and curl \(\mathbf{F}\). $$ \mathbf{F}(x, y, z)=\cos x \mathbf{i}+\sin y \mathbf{j}+3 \mathbf{k} $$

Find the center of mass of the homogeneous triangle with vertices \((a, 0,0),(0, b, 0)\), and \((0,0, c)\), where \(a, b\), and \(c\) are all positive.

Let \(\mathbf{F}=2 z \mathbf{i}+2 y \mathbf{k}\), and let \(\partial S\) be the intersection of the cylinder \(x^{2}+y^{2}=a y\) with the hemisphere \(z=\sqrt{a^{2}-x^{2}-y^{2}}\), \(a>0\). Assuming distances in meters and force in newtons, find the work done by the force \(\mathbf{F}\) in moving an object around \(\partial S\) in the counterclockwise direction as viewed from above.

In Problems 13-18, find div \(\mathbf{F}\) and curl \(\mathbf{F}\). $$ \mathbf{F}(x, y, z)=e^{x} \cos y \mathbf{i}+e^{x} \sin y \mathbf{j}+z \mathbf{k} $$

Show that the work done by a constant force \(\mathbf{F}\) in moving a body around a simple closed curve is 0 .

Find the mass of a wire with the shape of the curve \(y=x^{2}\) between \((-2,4)\) and \((2,4)\) if the density is given by \(\delta(x, y)=\) \(k|x|\).

Plot the parametric surface over the indicated domain. \(\mathbf{r}(u, v)=u \mathbf{i}+3 v \mathbf{j}+\left(4-u^{2}-v^{2}\right) \mathbf{k} ; 0 \leq u \leq 2\), \(0 \leq v \leq 1\)

A central force is one of the form \(\mathbf{F}=f(\|\mathbf{r}\|) \mathbf{r}\), where \(f\) has a continuous derivative (except possibly at \(\|\mathbf{r}\|=0\) ). Show that the work done by such a force in moving an object around a closed path that misses the origin is 0 .

In Problems 13-18, find div \(\mathbf{F}\) and curl \(\mathbf{F}\). $$ \mathbf{F}(x, y, z)=(y+z) \mathbf{i}+(x+z) \mathbf{j}+(x+y) \mathbf{k} $$

Plot the parametric surface over the indicated domain. \(\mathbf{r}(u, v)=2 u \mathbf{i}+3 v \mathbf{j}+\left(u^{2}+v^{2}\right) \mathbf{k} ;-1 \leq u \leq 1\), \(-2 \leq v \leq 2\)

A wire of constant density has the shape of the helix \(x=a \cos t, y=a \sin t, z=b t, 0 \leq t \leq 3 \pi\). Find its mass and center of mass.

Let \(S\) be a solid sphere (or any solid enclosed by a "nice" surface \(\partial S)\). Show that $$ \iint_{\partial S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S=0 $$ (a) By using Stokes's Theorem. (b) By using Gauss's Theorem. Hint: Show \(\operatorname{div}(\operatorname{curl} \mathbf{F})=0\).

Let \(f\) be a scalar field and \(\mathbf{F}\) a vector field. Indicate which of the following are scalar fields, vector fields, or meaningless. (a) \(\operatorname{div} f\) (b) grad \(f\) (c) curl \(\mathbf{F}\) (d) \(\operatorname{div}(\operatorname{grad} f)\) (e) curl \((\operatorname{grad} f)\) (f) \(\operatorname{grad}(\operatorname{div} \mathbf{F})\) (g) curl (curl F) (h) \(\operatorname{div}(\operatorname{div} \mathbf{F})\) (i) \(\operatorname{grad}(\operatorname{grad} f)\) (j) \(\operatorname{div}(\operatorname{curl}(\operatorname{grad} f))\) (k) \(\operatorname{curl}(\operatorname{div}(\operatorname{grad} f))\)

Calculate \(\iint_{2 S} \mathbf{F} \cdot \mathbf{n} d S\) for each of the following. Looked at the right way, all are quite easy and some are even trivial. (a) \(\mathbf{F}=(2 x+y z) \mathbf{i}+3 y \mathbf{j}+z^{2} \mathbf{k}\); \(S\) is the solid sphere \(x^{2}+y^{2}+z^{2} \leq 1\). (b) \(\mathbf{F}=\left(x^{2}+y^{2}+z^{2}\right)^{5 / 3}(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) ; S\) as in part (a). (c) \(\mathbf{F}=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}\) \(S\) is the solid sphere \((x-2)^{2}+y^{2}+z^{2} \leq 1\). (d) \(\mathbf{F}=x^{2} \mathbf{i} ; S\) is the cube \(0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\). (e) \(\mathbf{F}=(x+z) \mathbf{i}+(y+x) \mathbf{j}+(z+y) \mathbf{k} ; S\) is the tetrahedron cut from the first octant by the plane \(3 x+4 y+2 z=12\). (f) \(\mathbf{F}=x^{3} \mathbf{i}+y^{3} \mathbf{j}+z^{3} \mathbf{k} ; S\) as in part (a). (g) \(\mathbf{F}=(x \mathbf{i}+y \mathbf{j}) \ln \left(x^{2}+y^{2}\right)\); \(S\) is the solid cylinder \(x^{2}+y^{2} \leq 4,0 \leq z \leq 2\).

Let $$ \mathbf{F}=\frac{y}{x^{2}+y^{2}} \mathbf{i}-\frac{x}{x^{2}+y^{2}} \mathbf{j}=M \mathbf{i}+N \mathbf{j} $$ (a) Show that \(\partial N / \partial x=\partial M / \partial y\). (b) Show, by using the parametrization \(x=\cos t, y=\sin t\), that \(\oint_{C} M d x+N d y=-2 \pi\), where \(C\) is the unit circle. (c) Why doesn't this contradict Green's Theorem?

Plot the parametric surface over the indicated domain. \(\mathbf{r}(u, v)=2 \cos v \mathbf{i}+3 \sin v \mathbf{j}+u \mathbf{k} ;-6 \leq u \leq 6\), \(0 \leq v \leq 2 \pi\)

Find the work done by the force field \(\mathbf{F}\) in moving a particle along the curve \(C\). \(\mathbf{F}(x, y)=\left(x^{3}-y^{3}\right) \mathbf{i}+x y^{2} \mathbf{j} ; \quad C\) is the curve \(x=t^{2}\), \(y=t^{3},-1 \leq t \leq 0 .\)

Assuming that the required partial derivatives exist and are continuous, show that (a) \(\operatorname{div}(\operatorname{curl} \mathbf{F})=0\); (b) \(\operatorname{curl}(\operatorname{grad} f)=\mathbf{0}\); (c) \(\operatorname{div}(f \mathbf{F})=(f)(\operatorname{div} \mathbf{F})+(\operatorname{grad} f) \cdot \mathbf{F}\); (d) \(\operatorname{curl}(f \mathbf{F})=(f)(\operatorname{curl} \mathbf{F})+(\operatorname{grad} f) \times \mathbf{F}\).