Calculus and applications

Abstract

This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions $f:\mathbb{R}\to \mathbb{R}$, with the notion of continuity, and the construction of the derivative $f^{\prime }(x)$ and of the integral ${\int }_a^{b}f(x)dx$. Then we investigate the case of the complex functions $f:\mathbb{C}\to \mathbb{C}$, and notably the holomorphic functions, and harmonic functions. Then, we discuss the multivariable functions, $f:{\mathbb{R}}^N\to {\mathbb{R}}^M$ or $f:{\mathbb{R}}^N\to {\mathbb{C}}^M$ or $f:{\mathbb{C}}^N\to {\mathbb{C}}^M$, with general theory, integration results, maximization questions, and basic applications to physics.

400 pages