![]() ![]() Introduce $\tilde x=x$ because the partial derivatives $\partial_x g$ and $\partial_(1,2,3)$ hides step 1, taking for granted that the choice of restriction is obvious. If your coordinates are $x,y$ and you decided to introduce a new coordinate $\tilde y=x+y$, then you should also This consideration occurs in mechanics when the change of coordinates is introduced. For example, if $x=t^3$, then we differentiate $g$ with respect to $t$ holding ![]() You can treat the second computation as a partial derivative too it's just that instead of holding $x$ constant, we People don't bother introducing notation for it. Then we really look at the single-variable function $G(t):=g(x(t),t)$ but usually
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