# (Line integral) Compute work through vector field

1. May 7, 2013

1. The problem statement, all variables and given/known data
"Consider the Vector field F(x,y)=<cos(sin(x)+y)cos(x)+e^x, cos(sin(x)+y)+y>. Compute the work done as you traverse the Archimedes spiral (r=θ) from (x,y)=(0,0) to (x,y)=(2∏,0). (Hint: check to see if the vector field is conservative)

2. Relevant equations
1) F(x,y)=<P,Q> is conservative if $\partial$P/$\partial$y=$\partial$Q/$\partial$x

2) $\int$$_{c}$$\nabla$f$\cdot$dr= f(r(b))-f(r(a))

3. The attempt at a solution

1) The vector field is conservative by equation one: the partial derivative of P and Q with respect to y and x, respectively, are equivalent and equal cos(x)sin(sin(x)+y)

Difficulties:
The field is conservative, which means there exists a function f with ∇f=F. So I could use the fundamental theorem of line integrals, but I don't know how to integrate Q=cos(sin(x)+y)+y with respect to x.
As opposed to this I could try to do a change of variables but I don't know where to start with that.
I think some of my trouble comes from trying to wrap my head around r=θ, does that mean that the position vector r(x,y)=θ? or does it mean that r(r,θ)=<θ,θ>? If the latter is true, do I use polar coordinates? And if I do, how do I put F(x,y) into polar coordinates if r=<θ,θ>

2. May 7, 2013