TITLE: Change of variable for a smooth function
QUESTION [0 upvotes]: If we have a smooth function $f(x,y)$ and make a change of variables $x=u$ and $y=tu$, is the function $f(u,tu)$ smooth in $u $ and $t$? If yes, why?

REPLY [2 votes]: Let $g:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by
$$ g(u,t) = (u,tu)$$
As both function, $g_1(u,t) = u$ and $g_2(u,t) = ut$ are smooth functions (as they are polynomials), the function $g$ is smooth. Then if $f$ is smooth, $f\circ g$ is also smooth. Hence $f(u, tu) = f\circ g(u,t)$ is a smooth function.
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