How Do You Compute Work Done in a Vector Field with Polar Coordinates?

[naptor]

Homework Statement

The Problem states: Given the force vector field(in polar coordinates) : F(r,$\theta$)=-4Sin$\theta$i+4Sin$\theta$ j, compute the work done in moving a particle from (1,0) to the origin along the curve whose polar equation is : r=$e^{-\theta}$

The Attempt at a Solution

I converted the equation for the curve to Cartesian coordinates using (x,y)=rCos$\theta$i+rSin$\theta$i, however in order to find the line integral I think I should also have the vector in Cartesian, but I can't see a way to convert it. I basically did not understand the vector field equation it seems to have no relation whatsoever with r so ... Anybody has any tips how should I attack this one? and please, no spoilers unless it is just too easy

Homework Statement

 

[lippyka]

The Attempt at a SolutionIn order to solve this problem, we need to calculate the line integral of the force vector field F(r,θ) = -4sinθi + 4cosθj along the path of r = e−θ. To do this, we can first convert the polar equation for the curve into Cartesian coordinates by using the transformation (x,y) = rcosθi + rsinθj. This gives us the equation x = e−θcosθ and y = e−θsinθ. We can then use this equation to parametrize the curve in terms of t, giving us the expression (x(t), y(t)) = (e−tcos t, e−tsin t).Now, we can use this parametrization to calculate the line integral. We have:∫F•dr=∫F•(dx•i+dy•j)dtSubstituting in the expression for F and our parametrization, we have:∫F•dr=∫(−4sin t•ei−tsin t + 4cos t•e−tcos t)dtSolving this integral, we get the result that the work done in moving a particle from (1,0) to the origin along the curve is -8.