Best book for partial differential equations Which book is a self introductory for partial differential equations for beginners?
Good 1st PDE book for self study What is a good PDE book suitable for self study?   I'm looking for a book that doesn't require much prerequisite knowledge beyond undergraduate-level analysis.  My goal is to understand basic solutions techniques as well as some basic theory.
What is integration by parts, really? Integration by parts comes up a lot - for instance, it appears in the definition of a weak derivative / distributional derivative, or as a tool that one can use to turn information about higher derivatives of a function into information about an integral of that function. Concrete examples of this latter category include: proving that $f \in C^2(S^1)$ implies that the Fourier series of $f$ converges absolutely and uniformly, and the Taylor series expansion with the integral formula for remainder.  However, I don't feel like I really understand what integration by parts is really doing. To me, it is just an algebraic trick that follows from the fundamental theorem of calculus and the product rule. Is there some more conceptual way to think about it?  How do you think about this useful idea?