True or False The curl of a vector field $\mathbf{F}$ is a scalar.

If the vector field $\,\mathbf{F}=( e^{-x^{2}/2}-yz) \mathbf{i}+( e^{-y^{2}/2}+xz+2x) \mathbf{j}+( e^{-z^{2}/2}+5) \mathbf{k}$, then $\rm{curl}\, \mathbf{F}$=________.

Multiple Choice A vector field $\mathbf{F}$ is conservative if and only if $\rm{curl}\, \mathbf{F}=$ [(a) $\mathbf{F}$ , (b) div $\mathbf{F}$, (c) $\mathbf{0}$, (d) $\mathbf{n}$].

True or False An interpretation of the curl of $\mathbf{F}$ is circulation per unit area of a fluid at a given point on a surface.

Suppose $\mathbf{F}$ is the velocity vector of a fluid rotating about a fixed axis and that ${\bf\omega }$ is a constant angular velocity. Then $\rm{curl}\, \mathbf{F}=$________.

True or False If $\mathbf{F}$ is a conservative vector field, then the work done by $\mathbf{F}$ in moving an object of mass $m$ from point $A$ to point $B$ depends on the path taken from $A$ to $B$.

$\mathbf{F}(x,y,z)=x\mathbf{i}+y\mathbf{j}$

$\mathbf{F}(x,y,z)=y\mathbf{i}+x\mathbf{j}$

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

$\mathbf{F}(x,y,z)=4x \mathbf{i}-y\mathbf{j}-2z\mathbf{k}$

$\mathbf{F}(x,y,z)=3xyz^{2}\mathbf{i}+y^{2}\sin z\, \mathbf{j}+xe^{2z}\mathbf{k}$

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

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

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

$\mathbf{F}(x,y,z)=\cos x\mathbf{i}+\sin y\mathbf{j}+e^{xz}\mathbf{k}$

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

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

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

$\mathbf{F}=(z-y)\mathbf{i}+(z+x)\mathbf{j}-(x+y) \mathbf{k}$; $S$ is the portion of the paraboloid $z=1-x^{2}-y^{2},$ $z\geq 0$.

$\mathbf{F}=y\mathbf{i}+z\mathbf{j}+x\mathbf{k}$; $S$ is the portion of the paraboloid $z=1-x^{2}-y^{2}$, $z\geq 0$.

$\mathbf{F}=y\mathbf{i}-x\mathbf{j}$; $S$ is the hemisphere $z=\sqrt{1-x^{2}-y^{2}}$.

$\mathbf{F}=z\mathbf{i}+x\mathbf{j}+y\mathbf{k}$; $S$ is the hemisphere $z=\sqrt{1-x^{2}-y^{2}}$.

$\mathbf{F}=y^{2}\mathbf{i}+x\mathbf{j}-xz\mathbf{k}$; $S$ is the surface $z=1-x^{2}-y^{2}$, $z\geq 0$.

$\mathbf{F}(x,y,z)=4x\mathbf{i}-y\mathbf{j}+2z\mathbf{k}$; $S$ is the surface $z=1+x^{2}+y^{2}$, $z\leq 5$.

Use Stokes' Theorem to find the surface integral $\iint\limits_{\kern-3ptS}\hbox{curl}\,\mathbf{F} \,{\bf\cdot}\, \mathbf{n}\,dS,$ where $\mathbf{F}=\mathbf{F}( x,y,z) =y \mathbf{i}+x\mathbf{j}+x^{2}\mathbf{k}$ and $S$ is the surface enclosed by the paraboloid $z=9-$ $x^{2}-y^{2},$ $z\geq 0$.

Use Stokes' Theorem to find the surface integral $\iint\limits_{\kern-3ptS}\hbox{curl}\,\mathbf{F}\,{\bf\cdot}\, \mathbf{n}\,dS,$ where $\mathbf{F}=\mathbf{F}( x,y,z) =4z\mathbf{i }+3x\mathbf{j}+3y\mathbf{k}$ and $S$ is the surface enclosed by the paraboloid $z=10-x^{2}-y^{2},$ $z\geq 4$.

$\oint_{C}[(y+z)\,dx+(z+x)\,dy+(x+y)\,dz];$ $C$ is the curve of intersection of $x^{2}+y^{2}+z^{2}=1$, and $x+y+z=0$.

$\oint_{C}[(y-z)\,dx+(z-x)\,dy+(x-y)\,dz];$ $C$ is the curve of intersection of $x^{2}+y^{2}=1$, and $x+z=1$.

$\oint_{C}[x\,dx+(x+y)\,dy+(x+y+z)\,dz];$ $C$ is the curve $x=2\,\cos t$, $y=2\,\sin t$, $z=2$, $0\leq t\leq 2\pi$.

$\oint_{C}(y^{2}\,dx+z^{2}\,dy+x^{2}\,dz);$ $C$ is the triangle with vertices $(1,0,0),(0,1,0)$, and $(0,0,1)$.

$\mathbf{F}=x\mathbf{i}+y\mathbf{j}$

$\mathbf{F}=y\mathbf{i}+x\mathbf{j}$

$\mathbf{F}=xy\mathbf{i}+yz\mathbf{j}+zx\mathbf{k}$

$\mathbf{F}=yz\mathbf{i}+zx\mathbf{j}+xy\mathbf{k}$

Find the value of the constant $c$ so that $\mathbf{F}=xy \mathbf{i}+cx^{2}\mathbf{j}$ in space is a conservative vector field.

Find the value of the constant $c$ so that $\mathbf{F}=\dfrac{ z}{y}\mathbf{i}+c\dfrac{xz}{y^{2}}\mathbf{j}+\dfrac{x}{y}\mathbf{k}$, $y\neq 0$, is a conservative vector field.

Show that $\rm{curl}\,(\mathbf{F}+\mathbf{G})=\rm{curl}\, \mathbf{F}+\rm{curl}\,\mathbf{G}$.

Show that $\rm{curl}\,\left( c\mathbf{F}\right) =c(\rm{curl}\, \mathbf{F})$, where $c$ is a constant.

If $\ \mathbf{F}(x,y,z)=z\mathbf{i}+x\mathbf{j}+y\mathbf{k}$, find $\iint\limits_{\kern-3ptS}\mathbf{F}\,{\bf\cdot}\, \mathbf{n}\,dS$, where $S$ is the hemisphere $z=\sqrt{1-x^{2}-y^{2}}$.

Rework Problem 39, where $S$ is the circular region $x^{2}+y^{2}\leq 1$, $z=0$.

Show that $\mathbf{F}=y\mathbf{i}-x\mathbf{j}+z\mathbf{k}$ is not a conservative vector field. Nevertheless, there are certain paths $C$ for which $\oint_{C}\mathbf{F}\,{\bf\cdot}\, d\mathbf{r}=0$. Find one.

Show that div $\left( \rm{curl}\, \mathbf{F}\right) = \mathbf{0}$, where $\mathbf{F}=P( x,y,z) \mathbf{i}+ Q( x,y,z) \mathbf{j}+R( x,y,z) \mathbf{k}$ and $P$, $Q$, and $R$ are twice differentiable and the partial derivatives are continuous.

Suppose $\mathbf{F}(x,y,z)=P(x,y,z)\mathbf{i}+Q(x,y,z)\mathbf{ j}+R(x,y,z)\mathbf{k}$ is a vector field with continuous and differentiable components in a simply connected region of space.

Use Problem 43 to show that $\int_{C}[(yz-y-z)\,dx+ (xz-x-z) \,dy+(xy-x-y)\,dz]$ is independent of the path.

Work An object is moved from the origin to the point $(a,b,c)$ in the field of force $\mathbf{F}=(x+y)\mathbf{i}+(x-z)\mathbf{j} +(z-y)\mathbf{k}$. Show that the work done depends on only $a,b$, and $C$, and find this value.

Show that no twice differentiable vector function exists whose curl is $x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$.

Let $\mathbf{F}(x,y,z)$ be a vector field with continuous and differentiable components, and let $S$ be a sphere with outer unit normal $\mathbf{n}$. Use Stokes' Theorem to show that $\iint\limits_{\kern-3ptS}{\rm{curl}\, \mathbf{F}\,{\bf\cdot}\, \mathbf{n}}\,dS=0$.

Assume that the hypotheses of the Divergence Theorem hold for $S$ and $E.$ Show that for any vector field $\mathbf{F}$ with continuous first-order partial derivatives in a closed, bounded solid $E$ with orientable boundary $S$ with outer unit normal $\mathbf{n}$, $\ \iint\limits_{\kern-3ptS}{\rm{curl}\, \mathbf{F}\,{\bf\cdot}\, \mathbf{n}}\,dS=0.$

Let $C$ be a smooth, simple, closed curve lying on an orientable surface $S,$ and let $f$ and $g$ have continuous partial derivatives on $S$. Show that $\int_{C}(f{\bf\nabla }g)\,{\bf\cdot}\, d\mathbf{r} =\iint\limits_{\kern-3ptS}(\bf\nabla\ f\times {\bf\nabla }g)\,{\bf\cdot}\, \mathbf{n} \,dS$. Assume that the portion of $S$ bounded by $C$ is smooth and simply connected.