Why isn't Fourier Series taught in calculus 2? [closed]

Solution 1:

The primary use for Fourier series is solving second order differential equations which is not typically taught in Calculus II. Also the basic theory behind Fourier series is infinite dimensional vector spaces, certainly not taught in Calculus II!

Solution 2:

I am not sure what you mean by "Calculus 2", so here is a tentative answer.

For comparison, in France Fourier series are typically taught (at a basic level, up to Dirichlet's theorem and Parseval's theorem, in the second year of university. Taylor series are taught in first year (that would be, I guess, the first year of undergraduate studies in the United States). Differential equations are taught in first year (mainly through computation of solution of specific equations), while the general theorems (Cauchy-Lipschitz and similar) and series solutions (Frobenius method) are taught in second year. One would usually learn topology and the Lebesgue integral in third year (there is some topology before, but it's not even an overview, just the basic vocabulary).

The course on Fourier series is rather short compared to what is done on sequences and series of functions. They are both an application of series of functions, and the occasion to see prehilbertian structure and orthogonality in infinite dimension (but it's not greatly emphasized). Exercises are mainly about applications to the computations of numeric series.

Another answer seems to suggest infinite dimensional spaces are out of the question. They are actually seen (though very lightly) already in first year with spaces of polynomials and spaces of functions. However, the emphasis in first year is arguably on finite dimensional vector spaces and applications to matrices and geometry.

Caveat emptor: this is from remembrance of my academic studies 20 years ago, the curriculum may have changed a bit, and a teacher may have a different opinion.

Since you seem to be talking about high school (or secondary school): in France there is nothing about Rolle's theorem or Taylor series in high school. Calculus is limited to the computation of some limits, derivatives, and the basics about integration (integration by parts, but no change of variable). Differential equations are limited to 1st or 2nd degree linear ODE with constant coefficients. There is also some geometry and analytic geometry (up to conics and the use of complex numbers in plane geometry, and the basic theorems of trigonometry). I'd say the high school curriculum is rather light, compared to what is taught in university.

Note: if there is a reference for an up-to-date comparison of math curricula across countries, I would be greatly interested.