1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Vector calculas

  1. May 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Evaluate [tex]\int[/tex] F dr
    if F(x,y) =(6x2 +4y) i + (4x-2y) j and the curve C is a smooth curve from (1,1) to (2,3).

    2. Relevant equations

    3. The attempt at a solution
    I took partial derivatives with respect of y for the first term and with respect of x for the second term.
    4=4 => F is conservative, but from now I am confused how to solvet. I need at least the idea.

    I can find the patential function and the plug in the points values (1,1) and (2,3) for y and x .

    I am not sure if I can apply the Fundamental Thm of Line Integrals.

    I will appreciate any ideas how to finish the problem. Thank you.
  2. jcsd
  3. May 2, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper

    Hi mamma_mia66! :smile:
    Since you've proved F is conservative, that means you're free to choose any path …

    so just choose whatever looks easiest …

    I'd go for either (1,1) to (1,3) to (2,3), or (1,1) to (2,1) to (2,3) :wink:
  4. May 2, 2009 #3


    User Avatar
    Science Advisor

    Saying that Fdr is conservative means that there exist a function f(x,y) such that df= F ds and so the integral of Fdr between two points [itex](x_1,y_1)[/itex] and [itex](x_2,y_2)[/itex] is just [itex]f(x_2,y_2)- f(x_1,y_1)[/itex]

    So another way to do this is to find f(x,y) such that
    [tex]df= \frac{\partial f}{dx}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}= F dr[/tex]
    evaluated at (2,3) minus the value at (1, 1).
    That means you must have
    [tex]\frac{\partial f}{\partial x}= 6x^2+ 4y[/tex]
    [tex]\frac{\partial f}{\partial y}= 4x- 2y[/itex]
    From the first, f(x,y) must equal [itex]2x^2+ 4xy+ g(y)[/itex] since the "constant of integration" must depend on y only. Now, knowing that
    [tex]\frac{\partial f}{\partial y}= \frac{\partial (2x^2+ 4xy+ g(y)}{\partial y}= 4x+ \frac{dg}{dy}= 4x- 2y[/tex]
    what must f equal?
    Last edited by a moderator: May 2, 2009
  5. May 3, 2009 #4
    Thank you so much . I did finish the problem. I found the potential function and I pluged in the given points for y and x.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook