On Leibniz Notation

The derivative of 1 with respect to x is zero, since the derivative of any constant function is zero.

So you must be saying that the derivative of 1/f(x) with respect to x is zero for any f(x), where f is a differentiable function, with non-vanishing derivative near x (for it to be defined in the first place).

That doesn't make any sense.

Please don't respond to this. It's getting absurd.

This specific notation makes no sense in a Physics context. The names of the variables encode information that is lost if you use purely positional arguments. If I take a state function F(V, T) and its partial derivative wrt T, things like notation ∂_2 F get much murkier and context dependent than ∂_T F or ∂F/∂T, which benefit from a general consistency of notations. Ultimately, the function F has a physical meaning and the orders of the arguments is completely irrelevant, but what these arguments are is important. The Leibniz notation should certainly be improved or replaced, but not by this.

I mean, they even fuck it up in the sentence after their equation:

> The Lagrangian L is a real-valued function of time t, coordinates x, and velocities v; the value is L(t, x, v). Partial derivatives are indicated as derivatives of functions with respect to particular argument positions; ∂_2 L indicates the function obtained by taking the partial derivative of the Lagrangian function L with respect to the velocity argument position.

From what they say at the beginning the second argument is the position, not the velocity, so ∂_2 L is ∂_x L, not ∂_v L. Which is a really easy mistake to make because every single equation needs to go back to the definition of L to make sense.

> i've stumbled a few years ago on a math book that described the original concept of "infinitesimals" and how a whole different way of doing calculus exists.

Was this book "Elementary Calculus: An Infinitesimal Approach", by Keisler? It's an awesome book. It's free to download at https://people.math.wisc.edu/~hkeisler/calc.html

The choice of denominator letters are not irrelevant in Leibniz notation. They are chosen in conjunction with parameterization in order to suppress arguments to avoid being lengthy. Similar to how Einstein summation convention is used to suppress summation symbols. All notational conveniences lead to confusion in certain scenarios.

Given a field W, you will want to talk about various derivatives of the field. However, you might not be interested in the derivative of the fields with respect to coordinate fields like `\partial W / \partial x^i`, you might be interested in derivatives of that field with respect to some other field like `\partial W / \partial \lambda_i`. Then you end up introducing all kinds of auxiliary notation for the functional representations of the field. You end up with `\tilde{W}(\lambda_i)` and `\bar{W}(I_i)` next to `W(x_i)`.

Or just consider changes of variable, one generally doesn't distinguish the electric field `E` from cartesian, to axisymmetric, to spherical to whatever other coordinate system you dream up. Instead, the presence of the chosen coordinate symbols denotes the coordinate system in use and you don't need to distinguish the field from its functional representation. In order to understand `\partial_2 E` you would need to understand what coordinates I am working with while `\partial E / \partial y` is a bit more self explanatory in classical electrodynamics. Thermodynamics picks up confusion because of the constant changes in variable when the variables aren't coordinates.

> the concept of a function shouldn't depend on what your favourite letter is

On the other hand, you can also argue that the concept of a function shouldn't depend on your favourite ordering of its variables. Thus, if you have a potential that oscillates in time, such as:

    V = (1 + sin(t))/sqrt(x^2+y^2)

You may want to take derivatives with respect to each variable, such as dV/dt, dV/dx and so on. Their meaning is clear. What does D_2(V) mean, though? It does depend on your favourite ordering of the letters!

Even if you prefer positional notation for derivatives (à la Spivak), it must be recognized that the naming-based notation has its merits, and sometimes is clearer.