The Physics Travel Guide

Calculus

Best explanations

Intuitive

Calculus is the mathematics of quantities that change continuously. It has two halves, and each answers one practical question.

The first half, differentiation, asks how fast something changes at a given instant. Think of the speedometer in a car. The car's position changes all the time, and the speedometer shows how much distance the car would cover per hour if it kept its current pace. To get that number we look at the distance covered in a very short time interval and divide by the length of the interval. Calculus makes this precise by letting the interval shrink toward zero. The result is the derivative: the slope of the position-versus-time curve at one point.

The second half, integration, goes the other way. If we know the speed at every moment, how far has the car gone? We chop the trip into many short pieces, assume the speed is nearly constant in each piece, multiply speed by duration, and add up the pieces. As the pieces get shorter, the sum gets more accurate. In a graph of speed against time, this sum is the area under the curve.

The central result of the subject, the fundamental theorem of calculus, says that these two operations undo each other. Adding up small changes gives the total change, and the rate of change of an accumulated total is the thing being accumulated.

Physics needs this because its laws are mostly statements about rates of change. Newton's second law relates force to the rate of change of velocity. To predict where a planet or a pendulum will be later, we have to turn such statements about rates into statements about totals, and that is integration.

Concrete

Recommended Short Reads

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

Source: https://www.physicsforums.com/threads/any-calculus-starter-textbook-suggestions.905408/

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 ϵ\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. Grabiner, Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus, the American Mathematical Monthly, March 1983, Vol.90, No.3, 185-194.

See also https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/ and "The Cauchy-Schwarz Master Class"

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?

A Better Explained Guide To Calculus

Leads to: Differential Equations, Distributions, Exponential Function, Fourier Transform, Functional, Lagrange Multiplier, Laplace Transformation, Legendre Transformation, Leibniz Integration Formula, Manifold, Morse Theory, Newton's Second Law, Statistics, Variational Calculus, Vector Calculus.