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: Partial Derivatives

  1. Apr 8, 2009 #1
    1. The problem statement, all variables and given/known data

    Let [tex]u= (x^2 + y^2 + z^2)^\frac {-1} {2}[/tex]


    Find [tex]\frac {\partial^2 u} {\partial x^2} + \frac {\partial^2 u} {\partial y^2} + \frac {\partial^2 u} {\partial z^2}[/tex]

    2. Relevant equations
    3. The attempt at a solution

    [tex]\frac {\partial^2 u} {\partial x^2} = -(x^2 + y^2 +z^2)^\frac {-3} {2} + 3x^2(x^2 + y^2 + z^2)^\frac {-5} {2}[/tex]

    [tex]\frac {\partial^2 u} {\partial y^2} = -(x^2 + y^2 +z^2)^\frac {-3} {2} + 3y^2(x^2 + y^2 + z^2)^\frac {-5} {2}[/tex]

    [tex]\frac {\partial^2 u} {\partial z^2} = -(x^2 + y^2 +z^2)^\frac {-3} {2} + 3z^2(x^2 + y^2 + z^2)^\frac {-5} {2}[/tex]

    So I think all the partials are right, but I feel like I'm getting a crazy answer when I add them together.

    [tex]3x^2 + 3y^2 + 3z^2(x^2 + y^2 + z^2)^\frac {-5} {2} -3(x^2 + y^2 + z^2)^\frac {-3} {2}[/tex]

    Is this right?
     
  2. jcsd
  3. Apr 8, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Why do you think that's crazy? It looks correct to me. But you can express the answer in a much simpler form.
     
  4. Apr 8, 2009 #3
    here's the matlab quick code for the first step - finding d^2/dx^2:

    >> syms u; syms y; syms z;
    >> u = 1/sqrt(x^2+y^2+z^2)

    u =

    1/(x^2+y^2+z^2)^(1/2)


    >> diff(u,x)

    ans =

    -1/(x^2+y^2+z^2)^(3/2)*x


    >> diff(ans,x)

    ans =

    3/(x^2+y^2+z^2)^(5/2)*x^2-1/(x^2+y^2+z^2)^(3/2)
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook