Calculus and applications
Teo Banica
TL;DR
Calculus and Applications provides a guided introduction to calculus from foundational real-number constructs to core differential techniques. It develops limits, sequences, and series, builds intuition for continuity, and establishes the fundamental derivative rules (including chain, product, and quotient) along with pivotal results like the mean value and Rolle theorems. The text emphasizes concrete computations with powers, trigonometric, exponential, and logarithmic functions, and connects these ideas to broader analytic themes (uniform continuity, compactness, and inverse functions). By blending rigorous analysis with classical examples, it equips readers with tools for applying calculus to physics, probability, and beyond, while highlighting the construction and properties of important constants such as e. Overall, the work lays a solid foundation for both theoretical and applied aspects of calculus.
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.