The Fundamental Theorem For Line Integrals

(requires JavaScript)

1. Determine whether or not $\mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y\right)=⟨6x+5y,5x+4y⟩$ is a conservative vector field. If it is, find a function $f$ such that $\mathbf{F}=\nabla f$.
   $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=3x^2+5xy+2y^2+K$
2. Determine whether or not $\mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y\right)=x{e}^y\mathbf{i}+y{e}^x\mathbf{j}$ 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 $\mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\left(1+2xy+\ln <!--FUNCTION\; APPLICATION-->\left(x\right)\right)\mathbf{i}+x^2\mathbf{j}$ is a conservative vector field. If it is, find a function $f$ such that $\mathbf{F}=\nabla f$.
4. Let $\mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=yz\mathbf{i}+xz\mathbf{j}+\left(xy+2z\right)\mathbf{k}$ 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 ${\displaystyle {\int }_C\mathbf{F}•d\mathbf{r}}$.
   $f<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=xyz+z^2$ and $77$.
5. Show that the line integral ${\displaystyle {\int }_C\left(1-y{e}^{-x}\right)dx+e^{-x}dy}$ is independent of path and find its value along a path from $\left(0,1\right)$ to $\left(1,2\right)$.

6. Let ${\displaystyle \mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y\right)=P<!--FUNCTION\; APPLICATION-->\left(x,y\right)\mathbf{i}+Q<!--FUNCTION\; APPLICATION-->\left(x,y\right)\mathbf{j}=\frac{-y\mathbf{i}+x\mathbf{j}}{x^2+y^2}}$.

   Show that ${\displaystyle \frac{\partial P}{\partial y}}={\displaystyle \frac{\partial Q}{\partial x}}$, but ${\displaystyle {\int }_C\mathbf{F}•d\mathbf{r}}$ 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.)

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

      ${\displaystyle \mathbf{F}<!--FUNCTION\; APPLICATION-->\left(\mathbf{r}\right)=\frac{c\mathbf{r}}{{\left|\mathbf{r}\right|}^3}}$

      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, ${\displaystyle \mathbf{F}<!--FUNCTION\; APPLICATION-->\left(\mathbf{r}\right)=\frac{-mMG\mathbf{r}}{{\left|\mathbf{r}\right|}^3}}$. Find the work done by the gravitational field due to the Sun as the Earth moves from aphelion ($d_1=1.52\times {10}^8$ km) to perihelion ($d_2=1.47\times {10}^8$ km). Use values $m=5.97\times {10}^{24}$ kg, $M=1.99\times {10}^{30}$ kg, and $G=6.67\times {10}^{-11}$ $\mathrm{N}{\mathrm{m}}^2/{\mathrm{kg}}^2$.
   3. Let $\mathbf{F}$ be the electric force field, ${\displaystyle \mathbf{F}<!--FUNCTION\; APPLICATION-->\left(\mathbf{r}\right)=\frac{\epsilon qQ\mathbf{r}}{{\left|\mathbf{r}\right|}^3}}$. Suppose that an electron with a charge of $-1.6\times {10}^{-19}$ 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 $\epsilon =8.985\times {10}^9$.