- #1
qspeechc
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Hi.
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:
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.
So I'm reading a physics book and I come across the following passage:
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]
Ok, up to this point I'm fairly confident I'm following along. But then they do the following:
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]
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.
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