Toronto Math Forum
APM3462022S => APM346Lectures & Home Assignments => Chapter 2 => Topic started by: Yifei Hu on February 19, 2022, 08:10:02 PM

In text book 2.4, we had one line in the derivation: $\tilde{u_{\xi}}=\frac{1}{4c^2} \int^\xi f(\xi,\eta')d\eta'=\frac{1}{4c^2} \int_\xi^\eta\tilde f(\xi,\eta')d\eta' + \phi'(\xi)$.
Why can we replace the definite integral with the indefinite integral? Why we choose $\xi$ as lower limit and $\eta$ as upper limit?

Reproduce formula correctly (there are several errors) and think about explanation why $\phi'(\xi)=0$.

I understand that I can show $\phi'(\xi)=0$ by:
1) when t = 0, $\xi = x+ct = x = xct = \eta$
2) $u_\xi = u_t \frac{dt}{d\xi} + u_x \frac{dx}{d\xi}$ by chain rule.
3)By initial condition: $u_t_{t=0} = u_x_{t=0}$ = 0, we must have $u_\xi=0$
4) $u_\xi = \phi'(x)$ hence $\phi'(\xi)=0$
But how does this qualify us to replace the indefinite integral with the definite one?

But how does this qualify us to replace the indefinite integral with the definite one?
Did you take Calculus I? Then you must know that if the preimitive (indefinite integral) is a set of definite integrals which differ by an arbitrary constant.