Textbook in PDF format

This exploration of a selection of fundamental topics and general purpose tools provides a roadmap to undergraduate students who yearn for a deeper dive into many of the concepts and ideas they have been encountering in their classes whether their motivation is pure curiosity or preparation for graduate studies. The topics intersect a wide range of areas encompassing both pure and applied mathematics. The emphasis and style of the book are motivated by the goal of developing self-reliance and independent mathematical thought. Mathematics requires both intuition and common sense as well as rigorous, formal argumentation.
This book attempts to showcase both, simultaneously encouraging readers to develop their own insights and understanding and the adoption of proof writing skills. The most satisfying proofs/arguments are fully rigorous and completely intuitive at the same time.
Foreword
Sets and functions
Numbers
Vectors and their spaces
The topology of metric spaces
Differentiation
Manifolds
Ordinary differential equations
Optimization
Partial differential equations
Probability
11 Answers to selected questions