Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Last line integral problem (hopefully)

  1. Apr 30, 2005 #1
    Greetings again,

    Show that for F(x,y)=<2xy-3, x^(2)+4y^(3)+5> the line integral F(x,y).dr is independant of path. Then evaluate the line integral for any curve C with initial point (-1,2) and the terminal point (2,3).

    Thanks again, you all have been very helpful.
  2. jcsd
  3. Apr 30, 2005 #2
    has your class covered exact differentials?
  4. Apr 30, 2005 #3
    i don't believe so unless its known by another name?
  5. Apr 30, 2005 #4
    I'll post a detailed explanation tomorrow (if no one else has by then). I doubt you would know the subject by a different name.
    Last edited: Apr 30, 2005
  6. Apr 30, 2005 #5
    A line integral is independent of path if there exists a function U, that Fdr is it's exact (total) differential. In this case U=x^2y-3x+5y+y^4.
  7. Apr 30, 2005 #6


    User Avatar
    Science Advisor

    An "exact differential" fdx+ gdy (a physics major may prefer to think of it as a "conservative force field") is, as Oggy said, one such that there exist a function U such that dU= fdx+ gdy. By the "chain rule", [tex]dU= \frac{\partial U}{\partial x}dx+ \frac{\partial U}{/partial y}[/tex] so we must have [tex]f= \frac{\partial U}{/partial x}[/tex] and [tex]g= \frac{\partial U}{/partial g}. A quick way to test that is to use the "cross derivative test: If the second partials are continuous, that requires that
    [tex]\frac{\partial f}{/partial y}= U_{xy}= U_{yx}= \frac{\partial g}{\partial x}[/itex].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook