I initially planned to pen down this blog post after completing Michael Spivak’s Calculus. However, with my motivation dwindling near the end, I thought it best to start early. The primary impetus for delving deeper into mathematics is its indispensable role in comprehending science, and in turn, our world. Though I am less intrigued by the mathematical tool itself, its profound connection to science appeals to me.

I embarked on my self-learning journey on April 16, 2023. The book at the heart of this journey – Calculus by Michael Spivak – delves into single-variable calculus, elucidating topics through rigorous proofs, including limits, continuous functions, least upper bounds, intermediate value theorem, boundedness, extreme value theorem, derivatives, integrals, fundamental theorem of calculus, and the construction of various types of functions (Trigonometric Functions, Logarithm and Exponential Functions, etc). It took me about three months to navigate through 90% of the book, a task I found rather challenging. Through this blog, I aim to share some of my experiences and resources that I found useful.

Syllabus

My guide throughout this self-learning journey has been the syllabus of Math 185 – Honor Calculus I from the University of Michigan. This comprehensive syllabus covers one chapter per week, inclusive of a problem set. Though the pace was demanding, especially while juggling full-time work, I found it a fulfilling endeavour.

Additional online syllabi include:

• MthT 430 from the University of Illinois Chicago (Cover fewer materials)
• Math 113 from Johns Hopkins University

Learning Resources

Books

• Primary: Spivak, Michael. Calculus. Houston, Tex. : Publish or Perish, 2008. http://archive.org/details/calculus4thediti00mich.
• Supplementary (For Limits and Continuity):
  P. P. Korovkin. Limits And Continuity ( Pocket Mathematical Library), 1969. http://archive.org/details/korovkin-limits-and-continuity-pocket-mathematical-library.

Prerequisite

Before diving into this course, a basic understanding of proof-writing is highly recommended. I found the following resource quite useful in this context:

• Chartrand, Gary, Albert D. Polimeni, and Ping Zhang. Mathematical Proofs: A Transition to Advanced Mathematics. 2. ed., Pearson internat. ed. Boston: Pearson/Addison Wesley, 2008.

Though prior knowledge of inequality can prove beneficial, it isn’t a necessity. The first three chapters of the book acquaint you with the basic knowledge of inequality through problems at the end of the chapters. If you find the concept challenging, this book could be a useful reference:

• Kazarinoff, Nicholas D. Analytic Inequalities. Dover edition. Mineola, New York: Dover Publications, Inc, 2003.

Online Course

Although there are no accompanying online courses for Spivak’s Calculus, several courses use the renowned Real Analysis book: Rudin, Walter. Principles of Mathematical Analysis. One standout resource is the Real Analysis Lectures by Professor Francis Su (Harvey Mudd College), a former president of the Mathematical Association of America. His clear and engaging teaching style is particularly useful when grappling with topics like limits, continuity, derivatives, and the mean value theorem.

Asking For Help

As self-learners, we often lack dedicated assistance. However, the mathematical community is always at hand to offer help:

• Math StackExchange: For structured responses to well-formulated questions.
• Math Discord Channel: With over 156K members, this channel is ideal for discussions and more casual interactions.

Experiences

Staying motivated is the crux of self-learning. The absence of an external force pushing you makes it easy to succumb to discouragement. Why, you might ask, should you devote an extra 1-2 hours after a full 8-hour workday to studying Math? For me, the answer lies in my desire to comprehend the world more fully through the lenses of science and mathematics. Identifying your personal motivation is key.

The subject matter can indeed be daunting, especially when you first encounter the rigorous versions of limits and continuity. But rest assured, once you’ve crossed that hurdle, subsequent chapters become comparatively manageable (though still challenging). The problems posed are extremely challenging, often requiring hours spent on a single problem. However, don’t hesitate to seek help when stuck. The silver lining to these tough problems is the robust understanding they offer – far surpassing the insights gleaned from easier questions.

What’s Next?

Maybe a rigorous version of linear algebra?