Introduction to Calculus of Variations and The Fundamental Lemma

This is the first video of a series in mathematical physics. This series is intended for students who have completed the calculus series and know a little bit about proof writing. I intend for the series to be self-contained, but this may be difficult to do at times.

This video is a gentle introduction to calculus of variations and the fundamental lemma of the calculus of variations.

[BritishLions]

Very well explained, you have a talent for this!

[tombouie]

Well-done butttt, the proof by contradiction is well confusing.
Although it was quite the adventure, I just made it all the way thru the Euler-Lagrange eqn proof. All that's left is a clear proof of the basis of the calculus-of-variations.
A contradiction proof is a little like this: 1st here's the initial assumptions & long derivation, my conclusion are absolutely wrong, thus one or more of my initial assumptions must have been wrong, and ?are you confused enough yet?.
???Is there any clear proof of the basis of the calculus-of-variations that is clear/straight-forward????