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Math Help - partial derivatives - exercise

  1. #1
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    Red face partial derivatives - exercise

    From what I have heard this exercise should be easy, but I do not know how to resolve it .
    Any ideas?

    You have two functions, f(x,y) and g(x,y) which satisfy properties:
    df/dx = dg/dy and df/dy = - dg/dx.



    Show that,

    (d^2f)/(dx^2) + (d^2f)/(dy^2) = 0

    and

    (d^2g)/(dx^2)+(d^2g)/(dx^2) = 0

    Thanks!
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  2. #2
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    Quote Originally Posted by alekb View Post
    From what I have heard this exercise should be easy, but I do not know how to resolve it .
    Any ideas?

    You have two functions, f(x,y) and g(x,y) which satisfy properties:
    df/dx = dg/dy and df/dy = - dg/dx.



    Show that,

    (d^2f)/(dx^2) + (d^2f)/(dy^2) = 0

    and

    (d^2g)/(dx^2)+(d^2g)/(dx^2) = 0

    Thanks!

    So f_x=g_y\,,\,f_y=-g_x\Longrightarrow\,f_{xx}=g_{yx}\,,\,f_{yy}=-g_{xy} , and ASSUMING the second partial derivatives are continuous , or any other assumption that allows us to conclude that g_{xy}=g_{yx} , just sum the above and get zero.

    The second equality is very similar. Google "Cauchy-Riemann Equations" (or "conditions") and/or "harmonic functions".

    Tonio
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  3. #3
    MHF Contributor
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    Just in case a picture helps...



    ... where continuity of second partials gives...



    ... the given equations on the middle level imply...



    and



    ... the wanted results are...



    and



    ... and dashed and continuous lines differentiate downwards with respect to x and y or vice versa.


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