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Hi.

So I'm reading a physics book and I come across the following passage:

So I'm reading a physics book and I come across the following passage:

Ok, up to this point I'm fairly confident I'm following along. But then they do the following:Let [itex]p=(p_1,p_2,...,p_n)[/itex] and [itex]q=(q_1,q_2,...,q_n)[/itex] and the system of n algebraic equations

[itex]g_r(p,q)=0, \quad r\in \{1,2,...,n\}[/itex]

can be solved with respect to the [itex]p_i[/itex] in the form [itex]p_i=\phi _i(q)[/itex] for [itex]i\in \{1,2,...,n\}[/itex]

Then we have

[itex]\frac{d}{dq_h}g_r(\phi ,q)=0[/itex]

where [itex]\phi =(\phi _1,...,\phi _n)[/itex]. If we denote [itex]G_r(q)=g_r(\phi ,q)[/itex] then the last relation gives

[itex]0=\frac{\partial}{\partial q_h}G_s = \Big[\frac{\partial g_s}{\partial q_h} + \sum_{i=1}^{n}\frac{\partial g_s}{\partial p_i}\frac{\partial \phi _i}{\partial q_h} \Big]_{p=\phi (q)}[/itex]

and I have no idea where this comes from. I am guessing here that [itex]p_i=\phi _i(q)[/itex] is only in some sufficiently small region and not the entire space, since they do not explicitly say anything in this regard, so that [itex]p_i-\phi _i[/itex] is not identically zero on the whole space, so I do not know where the above comes from. This is probably something really simple and I'm just being stupid, but any help is greatly appreciated.Therefore the partial derivatives can be expressed in the form

[itex]\frac{\partial g_s}{\partial q_h} = \sum_{i=1}^n\frac{\partial g_s}{\partial p_i}\frac{\partial (p_i-\phi _i)}{\partial q_h}[/itex]

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