I am teaching myself differential geometry and mathematical physics from John Baez's book "Gauge Fields, Knots, and Gravity." I am getting hung up on one generic type of question: how to prove something is independent of coordinates or local trivialization, or that a mapping is well-defined. Here is an example:
Let $\pi:E\to M$ be a vector bundle. Let $\gamma:[0,T]\to M$ be a path on a smooth manifold. Let $u(t)$ be a vector in the fiber over $\gamma(t)$. Using the local trivialization $E\rvert_U\cong U\times V$, we define the covariant derivative as: $$D_{\gamma'(t)}u(t)=\frac{d}{dt}u(t)+ A(\gamma'(t))u(t)$$ where $A$ is an End$(V)$ valued function. Prove that this is independent of choice of local trivialization.
The book is full of other problems like this. I honestly have no clue where to start. In some sense it seems almost tautologically true: we are talking about algebraic objects for which we use coordinates/trivializations for ease of computation, but regardless of the representation they are always the same object. However, I do not know how to make this rigorous. I have tried arbitrary coordinate changes, but I usually end up with a mess of calculations that don't tell me what I need to know. In some other problems I have used theorems like the universal property of tensor products (e.g. exercise 83, if you happen to have a copy of the book), but in general I have just looked dumbfounded at the problem for a couple of hours before moving on.
Can anyone give me a step by step resolution of this problem? Or, even better, some sort of illumination on how to tackle these problems in general?
Thank you in advance!