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This set of anti gray boxes refer to file cm-lec-03008.pdf.
- [Eq(9)] A result that \[ \int f(x) dx \approx (b-a) f(x),\qquad \] requires some discussion.
- [Eq (21)]]We give some details here. Here you have an equation of the form
\begin{equation}\Label{A01} \int_{t_1}^{t_2} F_k(t) \delta q_k(t) dt =0. \end{equation}
where \(\delta q_k(t)\) are arbitrary and independent functions. You are required to prove \(F_k(t)=0 \) for all \(t_1< t< t_2\).
Since \(\delta q_k(t)\) are arbitrary and independent functions, you are allowed to choose \(\delta q_k(t)= F_k(t)\). This gives \begin{equation}\Label{A02} \int_{t_1}^{t_2} \sum_k \big|F_k(t)\big|^2 dt =0. \end{equation} Assuming \(F_k(t)\) to be continuous functions of \(t\), \eqRef{A02} holds if and only if \[ \sum_k \big|F_k(t)\big|^2 =0 \]
A sum of positive terms can be zero if and only if each term vanishes. This gives us (21). %
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