### The Fundamental Theorem For Line Integrals

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1. Determine whether or not $\mathbf{F}\left(x,y\right)=⟨6x+5y,5x+4y⟩$ is a conservative vector field. If it is, find a function $f$ such that $\mathbf{F}=\nabla f$.
$f\left(x,y\right)=3{x}^{2}+5xy+2{y}^{2}+K$
2. Determine whether or not $\mathbf{F}\left(x,y\right)=x{e}^{y}\mathbf{i}+y{e}^{x}\mathbf{j}$ is a conservative vector field. If it is, find a function $f$ such that $\mathbf{F}=\nabla f$.
Not conservative.
3. Determine whether or not $\mathbf{F}\left(x,y\right)=\left(1+2xy+\mathrm{ln}\left(x\right)\right)\mathbf{i}+{x}^{2}\mathbf{j}$ is a conservative vector field. If it is, find a function $f$ such that $\mathbf{F}=\nabla f$.
4. Let $\mathbf{F}\left(x,y,z\right)=yz\mathbf{i}+xz\mathbf{j}+\left(xy+2z\right)\mathbf{k}$ and let $C$ be the line segment from $\left(1,0,-2\right)$ to $\left(4,6,3\right)$. Find a function $f$ such that $\mathbf{F}=\nabla f$ and use it to evaluate ${\int }_{C}\mathbf{F}•\phantom{\rule{0.2em}{0ex}}d\mathbf{r}$.
$f\left(x,y,z\right)=xyz+{z}^{2}$ and $77$.
5. Show that the line integral ${\int }_{C}\left(1-y{e}^{-x}\right)\phantom{\rule{0.2em}{0ex}}dx+{e}^{-x}\phantom{\rule{0.2em}{0ex}}dy$ is independent of path and find its value along a path from $\left(0,1\right)$ to $\left(1,2\right)$.
6. Let $\mathbf{F}\left(x,y\right)=P\left(x,y\right)\mathbf{i}+Q\left(x,y\right)\mathbf{j}=\frac{-y\mathbf{i}+x\mathbf{j}}{{x}^{2}+{y}^{2}}$.

Show that $\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$, but ${\int }_{C}\mathbf{F}•\phantom{\rule{0.2em}{0ex}}d\mathbf{r}$ is not independent of path.

(Hint: Compute the integral along two different paths from $\left(1,0\right)$ to $\left(-1,0\right)$ along the unit circle.)

1. Let $\mathbf{F}$ be an inverse square force field:

$\mathbf{F}\left(\mathbf{r}\right)=\frac{c\mathbf{r}}{{\left|\mathbf{r}\right|}^{3}}$

for some constant $c$, where $\mathbf{r}=⟨x,y,z⟩$. Find the work done by $\mathbf{F}$ on an object which moves from a point ${P}_{1}$ to a point ${P}_{2}$ in terms of distances ${d}_{1}$ and ${d}_{2}$ from these points to the origin.

2. Let $\mathbf{F}$ be the gravitational force field, $\mathbf{F}\left(\mathbf{r}\right)=\frac{-mMG\mathbf{r}}{{\left|\mathbf{r}\right|}^{3}}$. Find the work done by the gravitational field due to the Sun as the Earth moves from aphelion (${d}_{1}=1.52×{10}^{8}$ km) to perihelion (${d}_{2}=1.47×{10}^{8}$ km). Use values $m=5.97×{10}^{24}$ kg, $M=1.99×{10}^{30}$ kg, and $G=6.67×{10}^{-11}$ $\mathrm{N}{\mathrm{m}}^{2}/{\mathrm{kg}}^{2}$.
3. Let $\mathbf{F}$ be the electric force field, $\mathbf{F}\left(\mathbf{r}\right)=\frac{\epsilon qQ\mathbf{r}}{{\left|\mathbf{r}\right|}^{3}}$. Suppose that an electron with a charge of $-1.6×{10}^{-19}$ C is located at the origin. Find the work done by the electric field due to the electron on a proton as the latter moves from the distance of ${10}^{-12}$ m from the electron to half that distance. Use the value of $\epsilon =8.985×{10}^{9}$.