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basic_tools:calculus

# Calculus

## Intuitive

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

## Concrete

Recommended Books:

• There is a small and fun to read book that makes the basics strong. The Hitchhiker's Guide to Calculus By Spivak
• For Applications, you cannot beat: No bullshit guide to math and physics by Savov
• Great way to quickly learn calculus Quick Calculus: A Self-Teaching Guide by Kleppner
• Great book (Q&A style) Calculus Refresher by Klaf
• A really fun book to read Zombies and Calculus by Adams
• Really good Calculus books: The Calculus Lifesaver by Banner and Calculus Made Easy by Thompson

A nice video introduction is

## Abstract

A more appropriate analogy would be that of calculus in its early days, more specifically when it was still plagued by infinitesimals { a very small quantity which is greater than zero yet less than any positive number, if you will. Sometimes we still think in this way, especially in physics (but this is because we already know that if we wish, we could always make it rigorous). The philosopher Berkeley was the first one to challenge the foundation of calculus. He remarked: They are neither finite quantities nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? It was due to criticism like this that finally led to rigorous formulation of calculus in terms of $\epsilon$ and δ now dreaded by beginning mathematics students [7] Note: Where is the Commutation Relation Hiding in the Path Integral Formulation? byYen Chin Ong

and ref 7 is: J. V. Grabiber, Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus, the American Mathematical Monthly, March 1983, Vol.90, No.3, 185-194.

## Why is it interesting?

We usually take shapes and formulas at face value, as a single pattern. Calculus gives us two superpowers to dig deeper:

X-Ray Vision: You see the hidden pieces inside a pattern. You don't just see the tree, you know it's made of rings, with another growing as we speak.

Time-Lapse Vision: You see the future path of an object laid out before you (cool, right?). "Hey, there's the moon. In the next few days it'll be rising and changing to a nice red color. I'll wait 6 days and take the perfect photo then."

So why is Calculus useful? Well, just imagine having X-Ray or Time-Lapse vision to use at will. That object over there, how was it put together? What will happen to it?