[johnnyb_61820]

If you are going to use infinitesimals, though, it requires some additional doing for notation. The standard notation for higher-order derivatives (and partial derivatives) needs to be modified in order for them to work (but they do work great once you do this).

Instead of the second derivative being "d^2y/dx^2" it is "(d^2y/dx^2) - (dy/dx)(d^2x/dx^2)" and the differentials can be manipulated just like any other entity. Additionally, you can infer this notation by simply applying the quotient rule to the first derivative (which is a quotient of infinitesimals).

See more:

"Extending the Algebraic Manipulability of Differentials" ( 10.48550/arXiv.1801.09553 )

"Total and Partial Differentials as Algebraically Manipulable Entities" ( 10.48550/arXiv.2210.07958 )

[Buttons840]

You're the author of this paper? Johnathan Bartlett?

If so, I used your calculus textbook to pass calculus at WGU. I had passed calculus in high school and university a long time ago, but when I finally decided to finish my degree I had to take it again, and got to choose my own text book; I liked your textbook best, I can see it sitting on my bookshelf right now.

https://www.amazon.com/Calculus-Ground-Jonathan-Laine-Bartle...

[johnnyb_61820]

Indeed! I'm glad you enjoyed the book! I hope you wrote it a nice Amazon review :)

[Buttons840]

I did. Glad to know you saw it.

[ianhorn]

On the topic, do you know any approaches to infitesimals/differentials that do cotangents and pullbacks as primitives?

In practice, I always end up needing to work in cotangents, but deriving them is always roundabout in terms of the limit definition of pushforwards. Never found a nice way to swap which is primary and which is secondary, but it feels like there should be a clean view of it that way somewhere.

[LegionMammal978]

How I like to think about it is that given an expression with a derivative dy/dx, we can always insert an arbitrary variable s that varies with both x and y, so that we can obtain an ordinary quotient (dy/ds)/(dx/ds) by the chain rule, and manipulate it normally with no qualms about what it means. As you say, second (and higher) derivatives can be calculated with the quotient rule.

[johnnyb_61820]

What I did in my book to keep everything algebraic but not introduce weird notation is just set the derivative equal to a variable. So, say m = dy/dx. Then, the second derivative is just dm/dx.

The advantage to the revised notation is that you can describe things that are difficult or impossible to describe in the other notation. For example, you can legitimately look at d^2y/d^2x (note the placement of the 2 on the denominator to see how this is different). This is a valid ratio under my system but invalid under the standard system (though I actually consider my system to be the standard system just with prior mistakes corrected).