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{\bf HW 9\\Math 261, F19}

Please see the course syllabus for details on how to turn in your homework assignments.  This one is due at the beginning of your class on \underbar{\bf Friday,  November 22}.


\begin{enumerate}
\item (5 pts.) \textsc{True or False:}
\begin{enumerate}
	\item If $f(x,y,z)$ is a differentiable function, then the gradient of $f$ is a conservative vector field.
	\item If $\mathbf{F}$ is a conservative vector field on a region $R$  and $C$ is a closed curve bounding $R$ then the line integral of  $\mathbf{F}$ along $C$ equals $0$.
	\item If $\mathbf{F}$ is a conservative vector field on $\mathbb{R}^3$ and $C_1$, $C_2$ two paths starting at a point $P$ and ending at a point $Q$ then the line integral of  $\mathbf{F}$ along $C_1$ equals the line integral of  $\mathbf{F}$ along $C_2$.
	\item If $\mathbf{F}$ is a conservative vector field, its potential function is unique.
	\item If $\mathbf{F}$ is a conservative vector field, its potential function is unique up to translation by a constant.
\end{enumerate}

\item (3pts.) Give an example of a conservative vector field, and of a vector field which is not conservative.

\item(3 pts.) Suppose conservative vector field $\mathbf{G}$ has potential function $g(x,y,z)=x^2+yz$.  Compute the work done when moving through this vector field along any simple curve from from $(0,1,1)$ to $(2,0,1)$.

\item (3 pts.) Find {\em the} potential function $f(x,y,z)$ for vector field $$\mathbf{F} = \langle \sin(y), x\cos(y)+z\cos(y), \sin(y) + 2z\rangle$$ such that $f(9,0,1)=2$.  You may assume that $\mathbf{F}$ is conservative.

\item (3 pts.) Use the component test ($M_y=N_x$, etc.) to show that the vector field
$$\mathbf{H}=\langle ze^{xz}-\sin(x+2y), \frac{1}{y} - 2\sin(x+2y) + 1, xe^{xz}+\frac{1}{z}\rangle$$
is conservative.  (Your solution should consist of three equalities.)

\item (3 pts.) Consider the vector field $\mathbf{F} = \langle y, -x, 0 \rangle$. Describe (or even better, sketch) how the vector field looks like for points of the unit circle in the $xy$- plane. Argue, without doing the component test, that the vector field cannot be conservative.


\end{enumerate}

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