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How to prove a function which is polynomial in the coordinates is differentiable everywhere

ianchenmu

Member
Feb 3, 2013
74
The question is:
Using the chain rule to prove that a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ which is polynomial in the coordinates is differentiable everywhere.



(The chain rule is for the use under function composition circumstances, how to apply it here to prove that the function $f$ which is polynomial in the coordinates is differentiable everywhere?)
 

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
(The chain rule is for the use under function composition circumstances, how to apply it here to prove that the function $f$ which is polynomial in the coordinates is differentiable everywhere?)
We can write:

$f:\mathbb{R}^n\to \mathbb{R},\quad f(x_1,\ldots,x_n)=\displaystyle\sum_{i_1\geq 0,\ldots,i_n\geq 0}a_{i_1,\ldots,i_n}x_1^{i_1}\ldots x_n^{i_n}$

with finitely many nonnull real coefficientes $a_{i_1,\ldots,i_n}$. Now, use the following properties:


$(1)\;$ The projections $\pi_i:\mathbb{R}^n\to\mathbb{R},\quad \pi_i(x_1,\ldots,x_n)=x_i$ are differentable on $\mathbb{R}^n$.

$(2)\;$ Every constant funcion is differentiable on $\mathbb{R}^n$.

$(3)\;$ The product of two differentiable functions on $\mathbb{R}^n$ is differentiable on $\mathbb{R}^n$.

$(4)\;$ The sum of two differentiable functions on $\mathbb{R}^n$ is differentiable on $\mathbb{R}^n$.
 

ianchenmu

Member
Feb 3, 2013
74
We can write:

$f:\mathbb{R}^n\to \mathbb{R},\quad f(x_1,\ldots,x_n)=\displaystyle\sum_{i_1\geq 0,\ldots,i_n\geq 0}a_{i_1,\ldots,i_n}x_1^{i_1}\ldots x_n^{i_n}$

with finitely many nonnull real coefficientes $a_{i_1,\ldots,i_n}$. Now, use the following properties:


$(1)\;$ The projections $\pi_i:\mathbb{R}^n\to\mathbb{R},\quad \pi_i(x_1,\ldots,x_n)=x_i$ are differentable on $\mathbb{R}^n$.

$(2)\;$ Every constant funcion is differentiable on $\mathbb{R}^n$.

$(3)\;$ The product of two differentiable functions on $\mathbb{R}^n$ is differentiable on $\mathbb{R}^n$.

$(4)\;$ The sum of two differentiable functions on $\mathbb{R}^n$ is differentiable on $\mathbb{R}^n$.
But where you used the chain rule? I mean, how to use the chain rule to prove that $f$ is differentiable everywhere?
 

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
But where you used the chain rule? I mean, how to use the chain rule to prove that $f$ is differentiable everywhere?
Perhaps an example will provide you the adequate outline. Consider $\pi_7(x_1,\ldots,x_n)=x_7$ and $g:\mathbb{R}\to \mathbb{R}$ given by $g(t)=t^3$. As $\pi_7$ and $g$ are differentiable, $(g\circ \pi_7)[(x_1,\ldots,x_n)]=x_7^3$ is also differentiable, etc, etc,...