What technique shows that a square wave is made up of a sine wave and its odd harmonics?
Why Fourier analysis is the mathematical technique that decomposes any repetitive waveform into a sum of sine waves: a fundamental plus harmonics at integer multiples of that frequency. For an ideal square wave the result is the fundamental plus only the odd harmonics (3rd, 5th, 7th...), each with amplitude 1/n of the fundamental. That is why a square wave passed through a narrow filter comes out as a sine wave, and why digital signals with fast edges splatter energy across many harmonics.
Watch out Vector analysis deals with quantities having magnitude and direction, useful for phasors and fields but not for breaking a waveform into harmonic components; numerical and differential analysis are general math tools with no specific role here.
Fourier = frequency. Square wave = fundamental + odd harmonics only.