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

Integral Of Total Differential

  1. Sep 2, 2004 #1
    I have to found a potential. So, I don't know how to integrate this:
    F = (x+2y+4z)i + (2x-3y-z)j + (4x-y+2z)k (bold are vectors)

    I think it goes like this:
    Potential...F = -grad U
    (x+2y+4z, 2x-3y-z, 4x-y+2z) = (-dU/dx, -dU/dy, -dU/dz)
    (-dU/dx,-dU/dy,-dU/dz are partial deriv.)


    -dU/dx=x+2y+4z
    -U=x^2/2+2y+4z+C(y,z)
    What know? :bugeye: Please help!
     
  2. jcsd
  3. Sep 2, 2004 #2

    Galileo

    User Avatar
    Science Advisor
    Homework Helper

    You've integrated -dU/dx with respect to x. The idea's good, but you did not integrate correctly.

    You know the x dependence of U and you know dU/dy and dU/dz as well.
    This is enough info to find U up to a constant.

    Good luck.
     
  4. Sep 2, 2004 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    dU/dx= x+2y+4z so U= (1/2)x<sup>2</sup>+ 2xy+ 4xz+ g(y,z)

    Given that, dU/dy= 2x+ dg/dy= 2x-3y-z so dg/dy= -3y- z

    Must have g= -(3/2)y<sup>2</sup>- yz+ h(z) and so
    U= (1/2)x<sup>2</sup>+ 2xy+ 4xz+ (3/2)y<sup>2</sup>- yz+ h(z).

    Given that, dU/dz= 4x- y+ h'= 4x-y+2z so h'= 2z.

    Now, what is h(z) and what is U?
     
  5. Sep 2, 2004 #4
    Oh, yes..now I see my mistake
    U is Potential.
    So, the result is:
    U = -1/2*x^2-2xy-4xz+3/2*y^2+yz-2+C
    Thank you for your help!!!
     
  6. Sep 2, 2004 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    No, You don't see. You copied the part I gave you did the tiny part I left incorrectly. You differentiated h'= 2z rather than finding the anti-derivative.
     
  7. Sep 5, 2004 #6
    But this result is correct. Like Galileo said - I had a good idea but I didn't integrate correctly
     
  8. Sep 6, 2004 #7

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    No, the result is NOT correct.

    If U= -1/2*x^2-2xy-4xz+3/2*y^2+yz-2+C

    Then Uz= -4x+ 1. The k component of grad U would be -4x+1, not -4x+y-2z as you want.

    I said, earlier, "U= (1/2)x2+ 2xy+ 4xz+ (3/2)y2-- yz+ h(z)
    Given that, dU/dz= 4x- y+ h'= 4x-y+2z so h'= 2z.

    Now, what is h(z) and what is U?"

    If h'= 2z then z= z2+ C (since h depends only on z, C is a constant.

    Then U(x,y,z)= (1/2)x<sup>2</sup>+ 2xy+ 4xz+ (3/2)y<sup>2</sup>- yz+ z2+ C.
     
    Last edited: Sep 6, 2004
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Integral Of Total Differential
  1. Total differential (Replies: 8)

  2. Total Differentials (Replies: 1)

Loading...