I have met two different definitions for Chern classes and I am wondering how are they all the same?
The first one is from Jöran Schlömer's book and they define the Chern class of a complex vector bundle $E \to M$ via the coefficients of $\det\left(tI - \frac{1}{2\pi i}A\right)= \sum_{k=0}^n f_k(A)t^{n-k}$ by setting $$c_k(E) = [f_k(\Omega)]$$ where $\Omega$ is the curvature matrix. The problem I have with this one is that I think it should be $$c_k(E) = \left[f_k\left(\frac{1}{2\pi i} \Omega\right)\right]$$ instead, but I'm no longer entirely sure.
The second definition is by Tu and he states that the Chern classes are obtained from $$\det\left(I + \frac{i}{2\pi}\Omega\right)=1+c_1(E)+\dots+c_n(E).$$
This I think is the total Chern class instead? If anyone could elaborate on why these two should give me back the same thing and whether or not it should be $$c_k(E) = [f_k(\Omega)]$$ or $$c_k(E) = \left[f_k\left(\frac{1}{2\pi i} \Omega\right)\right]$$ I would be glad. I think this might be due to some sign conventions.
The two formulas are exactly the same, with no sign difference.
For your concern of the first definition, the $\frac{1}{2\pi i}$ is contained in his definition of $f_k(A)$, so he didn't need to write it again.
Yes, the second one is the total Chern class. Just let $t=1$, you match it with the first one, note $$ \frac{i}{2\pi} = -\frac{1}{2\pi i}. $$