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February 16th, 2017, 02:47 AM  #1 
Member Joined: Jan 2016 From: United Kingdom Posts: 32 Thanks: 0  Multivariable Chain rule proof : is it correct?
Hello, I've had a go at proving the chain rule for a composition that maps from R^2 to R to R. There are two images attached. I've used the "arithmetic of limits" rules throughout, that if two limits exist, their product limit and sum limit exist etc. Any thoughts / scrutiny would be appreciated, since I can't find a proof elsewhere. James 
February 16th, 2017, 04:49 AM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 196 Thanks: 104 Math Focus: Dynamical systems, analytic function theory, numerics 
2 problems with your proof: 1. Look over your computations on page 1 (about halfway down). Consider if $f$ is constant and notice that none of these statements you have made are true. Namely, the RHS of (*) is not true when $f$ is constant. Unfortunately, this computation is the crux of your proof and the rest of the computation falls through. The difficult part of proving the chain theorem occurs precisely when $f$ has zero derivative on some sequence converging to $(a,b)$. 2. On page 2, you state from one line to the next that $$ \lim_{h \rightarrow 0 } \frac{g(f(a+h,b))g(f(a,b))}{f(a+h,b)  f(a,b)} = g'(f(a,b))$$ but this is precisely the statement that requires proof. Implicitly, in the line above you have interchanged the limit for the variable you call $t$ with the variable $h$. In this case that can be justified but this isn't always the case and any proof should argue why this is allowed here. As a hint toward a proof I suggest you consider the following expression $$ F(h) = \frac{g(f(a+h,b))  g(f(a,b))}{h}  g'(f(a,b)) \qquad h \neq 0 \\ F(0) = 0. $$ Now try to prove that $F$ is continuous at $h= 0$. Hope this helps. 
February 16th, 2017, 11:16 AM  #3 
Member Joined: Jan 2016 From: United Kingdom Posts: 32 Thanks: 0 
SDK , Thanks for the reply, I've attached an image to (try) to build on what you said about the constant f function. As for the rest, I'm going to assume the function is non constant and proceed with your idea. James 
February 16th, 2017, 11:28 AM  #4 
Member Joined: Jan 2016 From: United Kingdom Posts: 32 Thanks: 0 
Also, am I right in saying that to prove the continuity, one would have to use the fact that g is differentiable f(a,b)? Then manipulate the limit?


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