Remarks

Cross-cutting remarks on subtleties that span several of the mathematics pages — foundational fine print about how the rigorous machinery relates to the way mathematics is actually used, especially in physics.

Contents

1. Why the Physicist's Calculus Is Legitimate — why physics texts never define $\int fdx$ via $\sup /\inf$ (riemann-integral.md) and treat $dx$ as a length and $\frac{dy}{dx}$ as a fraction, yet always get the right answer: the integral half is Riemann sums with the limit hidden, the differential half is exact as an identity of 1-forms, infinitesimals are made literal by non-standard and smooth infinitesimal analysis, every informal move abbreviates a named theorem, and the hypotheses hold silently because physical fields are smooth — with a survey of the cases (uniform convergence, QFT divergences, $\delta (x)$, path integrals) where the shortcut genuinely bites.