In section 15.3 on page 379 there are numerous not-quite-correct assertions about Fourier series.
It repeatedly and emphatically asserts that «any» function can be represented by a Fourier series ... but this is not really true. Suggestion: It would be better to restrict attention to reasonably smooth, "well behaved" functions. (The formal necessary-and-sufficient conditions are beyond the scope of the course.)
It focuses on functions that are periodic in time, whereas the same ideas apply to all periodic waveforms, as a function of time, space, or whatever. Suggestion: Denote the period by P (not T).
It asserts that if we understand simple harmonic motion, we can deal with any other type of periodic motion by breaking it down into its simple harmonic components. However, in reality, that only works for _linear_ systems. In contrast, in nonlinear situations including atomic orbitals, shocks, solitons, spin waves, or even piano strings, you are free to Fourier analyze the waveform, but that is nowhere near sufficient for understanding the equations of motion. Suggestion: In this paragraph, or perhaps in this entire section, explicitly restrict attention to linear systems.
Most of the formal mathematical theorems of Fourier analysis involve infinite series. These theorems are sometimes useful for theoretical physics. A less formal but more broadly useful observation is that in a great many practical situations, the function can be well approximated by a finite Fourier series, involving only a few terms. Roughly speaking, this is analogous to approximating a function by a low-order polynomial. Suggestion: Mention this.
In section 15.3 on page 379 there are numerous not-quite-correct assertions about Fourier series.
if we understand simple harmonic motion, we can deal with any other type of periodic motion by breaking it down into its simple harmonic components.However, in reality, that only works for _linear_ systems. In contrast, in nonlinear situations including atomic orbitals, shocks, solitons, spin waves, or even piano strings, you are free to Fourier analyze the waveform, but that is nowhere near sufficient for understanding the equations of motion. Suggestion: In this paragraph, or perhaps in this entire section, explicitly restrict attention to linear systems.