...or: Plugging objects into an equation that it was absolutely not thought for.
Of course, all of you know the partial integration rule: $\int f' g = - \int f g'$ plus some boundary term we will impudently ignore. In your Analysis I course, both functions needed to be differentiable. But what if $f$ were piecewise differentiable? Perhaps even with jumps in the function values? Or unbounded, yet still integrable? In this talk, we will think about other ways to define $f'$ while keeping this equation valid. More precisely, we require nothing from real calculus, only this single equation.
Surprisingly, this requirement, all alone in the world, is not as lonely, lost and feeble as it seems, but already leads to a huge and beautiful theory (which could answer many questions from partial differential equations or calculus of variations, but that would be going too far). I will first introduce the most general notion for weak derivatives, the language of distributions. But you may recall your analysis courses on continuity or connectedness: What seems the most natural general definition is, in the domain of analysis, often a very nasty-behaving object. We will take a short look at the pitfalls we foolishly did not exclude in that first attempt and quickly move on to Sobolev spaces. As much as we can fit in half an hour, this will be the ultimate answer to life, universe and the Dirichlet problem.