Consider the vector field $F=-c \frac{x\mathbf{i}+y\mathbf{j}}{x^2+y^2}$. $$\mathbf{F}={-c}\frac{x\mathbf{i}+y\mathbf{j}}{x^2+y^2}$$ (vector field was rewritten here to make it easier to see) Consider the vector field above and using $c=1$, find by direct calculation the work done by the field in moving a unit mass along each of the following paths in the $xy$-plane. a. $C$ is the half line $y=1$, $x \geq 0$ b. C is the circle of radius a, with center at the origin, traced counterclockwise c. C is the line from (0,1) to (1.0) I actually have no idea where to start with this problem as i believe the professor never actually taught us this material but told us we would be assessed on it. I do believe that this is a line integral problem but I'm vexed as to how to set it up and solve it.

For all three first you will need to parameterize the path.

a) $x = t, y = 1$

b) $x = a\cos t, y = a\sin t$

c) $x = 1-t, y =t$

For each scenario find $\frac {dx}{dt}, \frac {dy}{dt}$

And with the appropriate intervals of $t$ for each path.

$\int_a^b \left(\frac {x(t)}{x(t)^2 + y(t)^2} \frac{dx}{dt} + \frac {y(t)}{x(t)^2 + y(t)^2} \frac{dy}{dt}\right) \ dt$

But, you might notice that $F$ is conservative (even though that is not being asked)

$F(x,y) = \
abla \left(-\frac 12 c \ln (x^2 + y^2)\right)$

And so, for all of these, we can just check the endpoints.