Example 1 (Chern form and Chern classes). Let $E$ be a complex vector bundle, and set
$$c(E,\nabla^E)=\det\left(\I+\frac{\sqrt{-1}}{2\pi}R^E\right) =\exp\left(\tr\left[\log\left(\I+\frac{\sqrt{-1}}{2\pi}R^E\right)\right]\right),$$
note that
$$\I+\frac{\sqrt{-1}}{2\pi}R^E$$
is invertible, and
$$\exp(\tr A)=\det(\exp(A)).$$
Note also that the power series
$$\begin{align*} \log(1+x)&=x+\frac{x^2}{2}+\cdots\\ \exp(x)&=1+x+\frac{x^2}{2!}+\cdots, \end{align*}$$
substitute $x$ with $\frac{\sqrt{-1}}{2\pi}R^E$, then it only has finite terms and this shows that for any integer $k\geq0$, $\tr[(R^E)^k]$ is a linear combination of various products of $c_i(E,\nabla^E)$’s, this established the fundamental importance of Chern classes in the theory of characteristic classes of complex vector bundles.

We write

$$\begin{align*} c(E,\nabla^E)&=\exp\left(\tr\left[\log\left(\I+\frac{\sqrt{-1}}{2\pi}R^E\right)\right]\right)\\ &=1+c_1(E,\nabla^E)t+c_2(E,\nabla^E)t^2+\cdots, \end{align*}$$
the $c_i(E,\nabla^E)$ are closed $2$-form, called Chern form, and $c(E,\nabla^E)\bigoplus c_i(E,\nabla^E)$ called the total Chern form, and the cohomology class associated to $k$-th Chern form $c_k(E,\nabla^E)$ are called the $k$-th Chern classes of second type, denoted as $c_k(E)$, and $\bigoplus c_k(E)$ are called total Chern classes.

Example 2 (Pontrjagin classes of real vector bundle). Let $E$ be a real vector bundle of $M$, define
$$p(E,\nabla^E)=\det\left(I-\left(\frac{\sqrt{-1}}{2\pi}\right)^2\right)^{\frac{1}{2}}.$$
Expand $\sqrt{1-x}$, we can write
$$p(E,\nabla^E)=1+P_1(E,\nabla^E)t+\cdots,$$
here $p_k(E,\nabla^E)$ is closed $4k$-form.