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Homework Help: (Line integral) Compute work through vector field

  1. May 7, 2013 #1
    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 [itex]\partial[/itex]P/[itex]\partial[/itex]y=[itex]\partial[/itex]Q/[itex]\partial[/itex]x

    2) [itex]\int[/itex][itex]_{c}[/itex][itex]\nabla[/itex]f[itex]\cdot[/itex]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)

    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. jcsd
  3. May 7, 2013 #2
    or do I need to put F(x,y) into one variable (say theta) and how do I do that? With r(x,y)=<θ,θ>, do I just plug that into F(x,y) or do I need to find f=antiderivative(F(x,y)) and then plug in r(x,y)=<θ,θ> for x and y
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