A lot of that is familiarity but I don't think all of it is.

Part of it is the brevity, and part of it is the shortcuts. E.g. when I did my masters, one thing I quickly realised was that papers that expressed an algorithm using mathematical notation almost always lacked essential details.

My impression is that it's too obvious when there are too large leaps in code, whereas in mathematical notation everyone accepts leaps that can obscure that essential details have been left out.

E.g. you'd have papers on thresholding of images for OCR (deciding what is background and what is foreground) where it turned out the results were highly dependent on certain values represented by certain variables that were never defined, for example, putting in a situation of reconstructing parameters by trial and error if you wanted to reproduce the results.

Today I'm immediately suspicious if results are presented as maths outside of fields where the maths is essential (and sometimes even then) as I see it as having a tendency to be used to gloss over sloppy work or save space by leaving out essential details.

I'm sure this is not the case in all fields, and that people with a more extensive maths background will be able to fill in more of those leaps without much effort, and so it might very well be acceptable in some fields. But to me a notation that makes it that easy to hide missing details is a liability.

I'd love to see this beauty or clarity or whatever that people find in mathematics, but I've never caught even a hint of it. Seems like it needs a good IDE to make up for deficiencies in its language.

That allows you to use Python inside a mathematical environment and it's amazing.

Saved my bacon a few times when I need to translate maths to code and I need to poke it a bit to get an understanding of how it works.

[0]http://www.sagemath.org/

(Linguists would want to murder me for saying this, I know.)

That's why some programming languages can even be defined by implementation. (Though as a programmer I try my best to avoid these languages...)

https://en.wikipedia.org/wiki/Total_derivative#The_total_der...

By the way, most of that article is terrible, but the linear map definition is the one we want. There's also the directional (Gâteaux) derivative:

https://en.wikipedia.org/wiki/Directional_derivative

Candidly, unless you really know your problem, we virtually always want the total derivative since it gives rise to things like gradients and Hessians, which are useful objects that we can store in memory.

Now, the reason that I bring these two up is that their spaces, or really their types, are different. Given a function f:X->Y, the total derivative is a linear operator from X to Y:

(total) f'(x) \in L(X,Y)

The directional derivative is an element in the space Y:

(dir) f'(x;dx) \in Y

Now, at this point, the notation is screwed up since we used Lagrange notation for both. The reason that we can get away with this is that under certain assumptions, that are mostly satisfied in the things we care about, we have that:

f'(x)dx = f'(x;dx)

Alright, so why should we care? Leibniz and Newton notation do a terrible job at capturing this information. Lagrange and Euler notation do a good job at this. For your example:

f(x0) = d/dx (sin(x)cos(x)+x^2) | x=x0

The types don't line up because sin(x)cos(x)+x^2 is value, literally a real number, not a function. Using the above, I would write this as:

(x \in R |-> sin(x)cos(x)+x^2)'(x0)

In LaTeX |-> would be \mapsto. This types correctly in the definitions above since

x \in R |-> sin(x)cos(x)+x^2 \in [R -> R]

and

(x \in R |-> sin(x)cos(x)+x^2)'(x0) \in L(R,R)

Of course, you probably wanted the value and not the function, which explains why we cheat in 1-D. So, we really should write:

(x \in R |-> sin(x)cos(x)+x^2)'(x0) 1 \in R

where we feed it the direction 1. And, yes, this is slightly more cumbersome that we may want, which is why there's a huge number of different notations. However, I do assert that the above generalizes properly all the way into infinite dimensions (Hilbert spaces) and provides a good foundation for typing out mathematical codes.

By the way, if anyone is looking for a book that does this right in my opinion, Rudin's "Principles of Mathematical Analysis" is amazing and his notation is good. For infinite dimensions, I prefer Zeidler's "Nonlinear Functional Analysis and Its Applications." Personally, what I look for is what I call properly typed notation that gives us easy access to useful tools like gradients, Taylor series, chain rule, and implicit and inverse function theorems. Again, most engineering and applied math work requires these theorems everywhere, so I find it best to keep them clean.

Is there any sensible rationale for sin^(-1)(x) meaning arcsin(x) while sin^2(x) means (sin(x))^2?

cf.

  f(x) = x^2 + 1
  x := 2
  f(2) = 5