Learning differential calculus through infinitesimals

Question

In class, we've studied differential calculus and integral calculus through limits. We reconstructed the concepts from scratch beginning by the definition of limits, licit operations, derivatives and then integrals. But the teacher really did everything to avoid talking about infinitessimals. For instance when we talked about variable changes we had to swallow that for a certain $u=\phi(x)$, $du = \phi'(x)dx$. What bothers me is how much we use them in maths and physics without understanding them. My question is how can I learn to manipulate infinitessimals? I've read an article on the field of infinitessimals but it didn't quite satisfy my curiosity. What literature would you recommend on the subject knowing that I have some background on linear algebra (vector spaces, matrices), algebraic structures (groups, rings, fields) and monovariable calculus.

Answer 1

Hope this relevant to what was asked...

To focus on $u = f(x), du = f'(x) dx,$ take this as an easy way to write $u = f(x), \Delta u = f'(x) \Delta x,$ where $\Delta x$ is an interval in $x$ that we can make arbitrarily small. This formula makes sense for $f$ smooth because in a small enough region around $x,$ $f$ is locally linear.

Recall that you can write $f(x) \approx f(x_0) + (x-x_0) f'(x_0) + {\cal O}((x - x_0)^2).$ Making $\Delta x \equiv x-x_0,$ and $\Delta f = f(x) - f(x_0),$ the original formula is obtained by ignoring the ${\cal O}(\Delta x^2)$. These definitions make sense because we have assumed that $f$ is smooth, so $\Delta f$ varies in proportion to $\Delta x$ (think of the delta-epsilon definition of continuity). None of this requires infinitesimals, it just requires continuity of the function and intervals that we can make "small enough". This is why multiplication by quantities like $dx$ makes sense.

Dealing with "actual" infinitesimals is the field of nonstandard analysis.

Answer 2

The usual treatment of calculus today avoids using infinitesimals entirely. $\mathrm{d}x$ is merely a notation that does not literally refer to an infinitesimal. That said, there is an alternative approach that uses infinitesimals known as nonstandard analysis. The Wikipedia articles on nonstandard analysis and nonstandard calculus are probably good places to start if you just want to find out if the topics interest you. Be aware these systems are not the usual treatment of calculus and people will be confused if you do not explicitly state you are working in them.

Answer 3

The best calculus textbook based on infinitesimals is Keisler's textbook Elementary Calculus. This opinion is based on our experience teaching calculus with infinitesimals based on the book over the past three years. We have taught over 250 students by now using this method, yielding better results than parallel groups that did not use infinitesimals, based on exam scores and teacher evaluations at the end of the course.