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!

The Fundamental Theorem for Line Integrals

  1. Nov 16, 2014 #1
    1. The problem statement, all variables and given/known data
    Determine whether or not f(x,y) is a conservative vector field.
    f(x,y) = <-3e^(-3x)sin(-3y),-3e^(-3x)cos(-3y) >

    If F is a conservative fector field find F = gradient of f

    2. Relevant equations
    N/A

    3. The attempt at a solution

    Fx = -3e^(-3x)(-3)cos(-3y)
    Fy = -3e^(-3x)(-3)cos(-3y)

    f is a conservative fector field
    (This part is all correct.)

    F = ??? + K

    It won't accept a vector, I know how to normally find a gradient vector, but that returns a vector, I need a non-vector answer..
     
    Last edited: Nov 16, 2014
  2. jcsd
  3. Nov 16, 2014 #2

    ShayanJ

    User Avatar
    Gold Member

    [itex]
    \vec \nabla f=<\partial_x f,\partial_y f>=\vec F=<F_x,F_y> \Rightarrow \left\{ \begin{array}{c} f=\int F_x dx \\ f=\int F_y dy \end{array} \right.
    [/itex]
     
  4. Nov 16, 2014 #3

    Zondrina

    User Avatar
    Homework Helper

    Since ##\vec F(x,y)## is conservative, you know ##\vec F(x,y) = \vec{\nabla f(x,y)}## for some potential function ##f##.

    This amounts to saying:

    ##\vec F(x,y) = \vec{\nabla f(x,y)}##
    ##P \hat i + Q \hat j = f_x \hat i + f_y \hat j##

    Equating the vector components: ##P = f_x## and ##Q = f_y##.

    So really you want to solve those two equations for some ##f(x,y)##. The usual method would be to integrate one of those two equations to find some ##f(x,y)## that has a constant of integration which varies. For example, taking the first:

    $$f(x,y) = \int P dx = P' + g(y)$$

    Afterwards, you should look at the equation you didn't use and observe the outcome.
     
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: The Fundamental Theorem for Line Integrals
Loading...