>>16654325
>I don't understand what your gripe is
Infinitesimals are already ill-defined for such a simple analytic structure as the reals. It’s all held together by sticks and play-doh. When you suddenly jump hoops and start talking about infinitesimals on much more complicated sets such as function spaces, you immediately open a whole can of worms. That’s why mathematicians don’t bother and do epsilon-delta stuff anyways.
> this guy already showed how to vary a functional along a single dimension, direction, etc
1. Not a single dimension or direction. It’s adding two functions together. Go ahead and make some plots of two sine waves added together and see how complicated and unintuitive it can get.
2. The way one proceeds is by Taylor expanding. This presents immediate questions:
a) You need to generalize Taylor expansions to deal with functions of multiple variables or even points on a manifold. Not as trivial as you think. There’s all kind of subtlety involved.
b) There’s an immediate assumption that the curves are analytic everywhere. And yet physicists have no gripes working with impulse functions that make the curves non-differentiable at isolated points.
c) Why do you drop terms of quadratic and higher order? What makes you justified in retaining linear terms but not quadratic terms? Because it’s “le smol”? Again, because we’re talking about functions, a function can take wildly different values at different points along the domain and a linear approximation may be valid at one point but not another. You need to carefully define radius of convergence via the epsilon-delta games etc.

I know I’m nitpicking, but it just grinds my autistic gears how physicists are so up their asses about these ad hoc derivations. If they were to admit that they’re doing reckless bullshit, then I’d give them slack. But it’s always this “it just works bro shut up” mentality instead.

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>be a physics student>study Lagrangian mechanics and field theory>textbooks and instructors use bullshit “infinitesimals” to derive…

>be a physics student
>study Lagrangian mechanics and field theory
>textbooks and instructors use bullshit “infinitesimals” to derive…