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Math Help - Partial Differentiation

  1. #1
    Super Member Aryth's Avatar
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    Partial Differentiation

    A function f(x,y,z) is called homogeneous of degree n if f(tx,ty,tz) = t^nf(x,y,z). For example, z^2\ln{\left(\frac{x}{y}\right)} is homogeneous of degree 2 since

    (tz)^2\ln{\left(\frac{tx}{ty}\right)} = t^2\left(x^2\ln{\frac{x}{y}}\right)

    Euler's theorem on homogeneous functions says that if f is homogeneous of degree n, then

    x\frac{\partial{f}}{\partial{x}} + y\frac{\partial{f}}{\partial{y}} + z\frac{\partial{f}}{\partial{z}} = nf

    Prove this theorem.

    Can someone just get me started?
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  2. #2
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    Quote Originally Posted by Aryth View Post
    A function f(x,y,z) is called homogeneous of degree n if f(tx,ty,tz) = t^nf(x,y,z). For example, z^2\ln{\left(\frac{x}{y}\right)} is homogeneous of degree 2 since

    (tz)^2\ln{\left(\frac{tx}{ty}\right)} = t^2\left(x^2\ln{\frac{x}{y}}\right)

    Euler's theorem on homogeneous functions says that if f is homogeneous of degree n, then

    x\frac{\partial{f}}{\partial{x}} + y\frac{\partial{f}}{\partial{y}} + z\frac{\partial{f}}{\partial{z}} = nf

    Prove this theorem.

    Can someone just get me started?
    Differentiate both sides of the equation f(tx,ty,tz) = t^nf(x,y,z) with respect to t, using the chain rule. Then put t=1.
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