1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

(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)

    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. 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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: (Line integral) Compute work through vector field
Loading...