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Homework Help: Second order partial derivative

  1. Dec 11, 2009 #1
    1. The problem statement, all variables and given/known data
    a and b are functions of z:

    a=a(z); b=b(z)

    I want to calculate the second order partial derivative operator on z
    2. Relevant equations
    Using the chain rule:

    [tex]
    \frac{\partial}{\partial z}=\frac{\partial a}{\partial z}\frac{\partial}{\partial a}+\frac{\partial b}{\partial z}\frac{\partial}{\partial b}
    [/tex]
    3. The attempt at a solution
    Is it correct?
    [tex]
    \frac{\partial^2}{\partial z^2}=\frac{\partial}{\partial a^2}+2\frac{\partial^2}{\partial a\partial b}+\frac{\partial^2}{\partial b^2}
    [/tex]
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Dec 11, 2009
  2. jcsd
  3. Dec 11, 2009 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Your problem is incompletely stated and uses poor notation and I think the answer is no anyway. It would be much clearer to use subscript notation for partial derivatives and ' for ordinary derivatives. Here is what I am guessing you are asking.

    Let w = f(a,b) where a and b are functions of z. Differentiating with respect to z, the derivatives of a and b would be a' and b', not partial derivatives. The derivatives of w with respect to a and b would be partials: wa and wb. The derivative of w with respect to z would be w'. The chain rule gives:

    w' = faa' + fbb'

    This is your chain rule you have given in the relevant equations. Now you want to calculate w''.

    w'' = (faa' + fbb')'

    Can you take it from there?

    = (fa)'a' + faa'' + (fb)'b' + fbb''
     
  4. Dec 12, 2009 #3

    HallsofIvy

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    Science Advisor

    Do you mean that you want to think of z as a function of a and b and find
    [tex]\frac{\partial^2 z}{\partial a^2}[/tex]
    [tex]\frac{\partial^2 z}{\partial b^2}[/tex]
    and
    [tex]\frac{\partial^2 z}{\partial a\partial b}[/tex]?

    This makes no sense at all. What function are you differentiating?

     
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