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Math Help - Chain rule with multiple variables

  1. #1
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    Chain rule with multiple variables

    My real analysis textbook asks me to

    Let h(u,v)=f(u+v,u-v). Show that f_{xx}-f_{yy}=h_{uv}.

    It looks like a fairly simple problem, at the end of the chapter on the chain rule. However, when I apply the chain rule, I get

    h_{uv}=f_{xx}-f_{yy}+f_{yx}-f_{xy}

    Now, if I can show that either f_{xy} or f_{yx} is continuous on \mathbb{R}, then I can let f_{xy}=f_{yx} and the conclusion follows. But how do I prove that with so little information? Or am I going about this entirely the wrong way?

    Thanks!
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  2. #2
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    Quote Originally Posted by hatsoff View Post
    My real analysis textbook asks me to

    Let h(u,v)=f(u+v,u-v). Show that f_{xx}-f_{yy}=h_{uv}.

    It looks like a fairly simple problem, at the end of the chapter on the chain rule. However, when I apply the chain rule, I get

    h_{uv}=f_{xx}-f_{yy}+f_{yx}-f_{xy}

    Now, if I can show that either f_{xy} or f_{yx} is continuous on \mathbb{R}, then I can let f_{xy}=f_{yx} and the conclusion follows. But how do I prove that with so little information? Or am I going about this entirely the wrong way?

    Thanks!
    What is the exact statement of the problem? What you have written makes no sense. For example, there is nothing said, in the hypothesis, about u and v being functions of x and y! Was there something said about f(x,y)?
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    What is the exact statement of the problem? What you have written makes no sense. For example, there is nothing said, in the hypothesis, about u and v being functions of x and y! Was there something said about f(x,y)?
    What I wrote in my OP is the original wording. I assume that since this is a textbook on real analysis, we have f:\mathbb{R}^2\to\mathbb{R} and that x=u+v and y=u-v, where (u,v)\in\mathbb{R}^2, but perhaps my assumptions are incomplete (or even wrong).

    If you doubt me, the textbook is freely available from the author in pdf format, here.

    See exercise 12a from section 5.4, p358---or pdf page 367.
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