Calculus and applications

TL;DR

This paper provides an introductory overview of calculus, covering real, complex, and multivariable functions, their derivatives, integrals, and applications to physics, emphasizing foundational concepts and theories.

Contribution

It offers a comprehensive introduction to calculus across different types of functions and explores their applications in physics, integrating theory with practical insights.

Findings

Fundamental concepts of derivatives and integrals for real functions

Extension of calculus to complex and multivariable functions

Applications of calculus to basic physics problems

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 $f'(x)$ and of the integral $\int_a^bf(x)dx$. 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.