During my physics PhD, I had to use the Variational or 'Weak' Form of a PDE, which derives from the Calculus of Variations. Since I had never encountered any of this in my class-based education, I had to teach myself, using whatever resources I could find on the web. All the stuff I found was clearly aimed at mathematicians - it was very terse, and got the motivating physics wrong. When I finally had a grip on the maths, I wrote a fairly detailed exposition of the calculus of variations, and included it as an appendix to my thesis.
This is what I would have wanted to learn from when I started: it is aimed at physicists, so no large leaps in mathematical reasoning are made, and the motivating physics is not confusingly wrong. Other explorers may find it useful, so I have compiled the appendix (with Latex) as a standalone chapter, and made the PDF available here:
Calculus of Variations and the Variational or 'Weak' Form of PDEs
It may just be the easiest explanation of calculus of variations (and the variational or 'weak' form) on the internet.
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