Machine Learning

Appendix C

Calculus of Variations

When I was in high school, my physics teacher

 — whose name was Mr. Bader — called me down one day

after physics class and said,

“you look bored; I want to tell you something interesting.”

Then he told me something which I found absolutely

fascinating, and have, since then, always found fascinating.

Every time the subject comes up, I work on it.

Richard Feynman

C.1 Functionals and Variations

Let us start describing functionals and related classic problems by some examples.

Example 1

Let $f\in \mathcal{F}:=C^1(A\subset \mathbb{R},B\subset \mathbb{R})$ be and consider

(C.1.1) ${\mathcal{L}}_1(f)=\underset{x_1}{\overset{x_2}{\int }}\sqrt{1+{(y^{\prime }(x))}^2}dx$

(C.1.1)

where $\forall x\in A:y(x)=f$