I think I have a problem with a basic understanding of partial derivatives when changing variables.
I do understand that if we just change variables, e.g. $(x_1, x_2, x_3) \rightarrow (y_1,y_2,y_3)$, the partial derivatives change as $\frac{\partial}{\partial x_i} \rightarrow \sum\limits_{j=1}^{3}\frac{\partial y_j}{\partial x_i}\frac{\partial }{\partial y_j}$. However, I do not understand what happens if we reduce the number of variables by introducing constraints. For example, suppose that equation $g(x_1,x_2,x_3)=0$ is fulfilled. Now we have 2 degrees of freedom left, so how does one calculate the partial derivatives with respect to the new variable. For concreteness, let's assume that the new variables are $(x_1, x_2)$ and $x_3 = f(x_1, x_2)$, so what are $\frac{\partial}{\partial x_1}$ and $\frac{\partial}{\partial x_2}$?
If you have a function $h(x,y,z)$, and you define $z=f(x,y)$, then the chain rule says
$$ \frac{\partial}{\partial x} h(x,y,f(x,y)) = \frac{\partial h}{\partial x} + \frac{\partial h}{\partial z} \frac{\partial f}{\partial x} $$
and similarly for $y$. In this formula, the terms $\frac{\partial h}{\partial x}$ and $\frac{\partial h}{\partial z}$ on the right-hand side should be understood to be evaluated at $z=f(x,y)$.