Why isn't Fourier Series taught in calculus 2?

Question

I am self taught and have read the book "Essential Calculus ETF" by Larson and Hosteller from cover to cover and have since been evaluating the more difficult problems in calculus. During my venture in solving difficult integrals using series expansions via the monotone convergence theorem, I noticed that some integrals were tackled easily with the Fourier series.

Now, why isn't this topic included in the same sections where power, taylor and maclaurin series are being discussed?

Not that I'm complaining (though I favor maclaurin and this is my hobby, so I'm learning something new), but shouldn't this series also be included in Calculus 2?

Answer 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!

Answer 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.

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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.

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

Answer 3

Not that this is entirely new, but I'd like to put my $2-$cents in.

Like most of mathematics, you can learn the basics of any rich theory at any time. However, to truly understand something, and to see it's utility and purpose, you may need to understand other areas of mathematics.

Fourier series aren't discussed in Calculus II, because they would be unmotivated. Why exactly should $f(x)$ be representable as an infinite series of integrals, using sine and cosine? The answer lies in the fact that the integral is a useful way of defining an inner-product on a vector space of continuous functions. This is useful in that it allows you to define the angle between two functions, and it allows you to associate a length to a function. The integrals you compute are giving you vector building blocks which approximate your function, but the why is because these variants of sine and cosine produce what is called an orthonormal basis (whatever that means).

Fourier series is a powerful tool, which would be difficult to convey without the language of linear algebra, which typically taught after Calculus II and before Differential Equations.

At my institution, we teach Fourier series right after vector calculus. When students have a sufficient understanding of linear algebra to understand why Fourier series should work.