I have no problem with whatever coordinates you use until you get the right answer, but if you're struggling with a simple computation stick to normal people methods. $\|\alpha(t)\|^2 = x(t)^2 + y(t)^2 + z(t)^2$.
Differentiate that, you get $2(x(t)x'(t) + y(t)y'(t) + z(t)z'(t))$.
That's $2\alpha(t) \cdot \alpha'(t)$.
But $\|\alpha(t)\|^2$ is minimized at $t = t_0$, so the derivative at that point is zero. That in turn implies $\alpha(t_0) \cdot \alpha'(t_0) = 0$.
The high points in multivariable calculus are the Jacobian formalism, second derivative test, inverse/implicit function theorem and change of variables theorem in my opinion
Basically if you have full rank $Df(x_0)$, then to prove surjectivity of $f$ in a neighborhood of $x_0$ is given by solving for $f(x) = y$ for some given $y$ in the codomain neighborhood of $f(x_0)$.
@BalarkaSen OTOH, most people prove the area formula for Hausdorff measure and then notice it's the change of variables formula in a special case. The proof of that is nontrivial, although the basic idea is the same
You need these things called Federer approximations
@BalarkaSen I don't know how people can get invested in GTM as a career. The high level ideas are nice, granted, but the details...
@0ßelö7 I don't really know how I feel about it - it was so long ago, my brain's very, very dusty on it, but at the same time, it was a fair amount of time and effort put into it, so it now feels like a bit of a waste :/
and yet when I think about spacetime intervals, my knee jerk reaction is to have timelike separations be positive and spacelike separations be negative.
in special relativity, neither distances nor times are invariant quantities. however, $ds^2=dx^2+dy^2+dz^2-c^2 dt^2$ is invariant under Lorentz transformations
so one way to make it work is to pretend that the photon does have a finite mass, and then take the limit at the end of the calculation to see what you get
and this is equivalent to changing $f(t)$ a little bit so that the poles move off the real axis.
@BalarkaSen There was a post on PSE where someone argued existence/uniqueness for a bunch of PDEs based on the fact that they represent physical systems
So just set up the physical system and there's your solution
@0ßelö7 Oh speaking of I heard some sketchy thing about that if you have a nonlinear ODE $x' = f(x)$ w/ initial condition $x(0) = 0$ say and you pull a first order Taylor to get the new system $x' = Ax$ where $A$ is the derivative, if $A$ is hyperbolic then these two systems are semiconjugate or something
it's only when ask how they behave under transformations, specifically spatial inversions, that the distinction between vector and pseudo-vector matters
