Exploring Mathematics Beyond High School: A Comprehensive Guide

Mathematics is a vast field that extends far beyond the scope of high school education. While a significant portion of advanced mathematical concepts are introduced in college, there are many specialized areas that are typically reserved for graduate studies. For top-performing students, these might be covered in their final undergraduate years. High schools generally focus on teaching Calculus up to the AP AB and BC level, which might take two years. This is a good introduction to the fundamental concepts like derivatives, integrals, and infinite series, preparing students for more advanced mathematics at the university level.

Advanced Mathematics Beyond College

After the basic Calculus courses, college students often delve into more specialized areas. These include:

Partial Differential Equations: These equations are crucial in many fields, including physics and engineering. Complex Analysis: This branch of mathematics deals with complex functions and their properties, often covered alongside Real Analysis and Partial Differential Equations. Contour Integrals: These integrals are a fundamental component of Complex Analysis, involving integration over a closed curve in the complex plane. Infinite Series and Sequences: Advanced studies of infinite series and sequences, including convergence tests and advanced integrals. Taylor and Laurent Series: These series expansions are used to represent functions in a series and are often used in complex analysis and differential equations. Fourier and Laplace Transforms: These mathematical tools are used to solve differential equations and analyze functions in various applications. Differential Geometry: This branch deals with the study of geometric properties of curves and surfaces in higher dimensions.

My High School and College Mathematics Experience

My own journey in learning mathematics began in college, as I did not have the opportunity to study Calculus in high school. However, I made up for lost time by taking the following courses:

Year One

Two semesters of Calculus One semester each of Logic and Differential Equations

Year Two

Two more semesters of Calculus

Year Three

Two semesters of Advanced Calculus Two math and computer science classes, including Linear Algebra and Topology

Year Four

Graduate classes in Point Set Topology and Algebraic Topology Two semesters of Complex Analysis Two semesters of Modern Algebra Two semesters of Differential Geometry One semester of the History of Mathematics

While this list only covers 24 or 25 classes, it gives you an idea of the depth and breadth of mathematics beyond the high school level.

Self-Learning Mathematics with Classic Texts

For those looking to get started with advanced mathematics, especially calculus, there are a number of resources available. One excellent book is Calculus Made Easy by Sylvanus P. Thompson, originally published over 100 years ago and still in print today. Updated versions by noted math writer Martin Gardner have also been produced. The book is known for its clear and accessible explanations, making it a great tool for self-study.

Even the original Thompson edition is now in the public domain and available for free online. It is highly recommended for anyone looking to learn calculus on their own. Keep in mind that if you order the Kindle version, you might get the original Thompson edition, so be sure to check the version you are purchasing.