= Solution
A <connection on a vector bundle> is a complex-linear map from smooth sections to vector-bundle-valued one-forms satisfying
$$
D(fs)=df\otimes s+fDs.
$$
Choose a locally finite trivializing cover, its componentwise flat <connections on a vector bundle> $D_i$, and a subordinate smooth <partition of unity> $\rho_i$. The formula $D=\sum_i\rho_iD_i$ is globally meaningful: each weighted term extends by zero outside its chart. Its <Leibniz rule> follows from $\sum_i\rho_i=1$, proving \b[every smooth complex vector bundle admits a connection].
Extend the <connection on a vector bundle> to bundle-valued <differential forms> by
$$
D(\beta\otimes s)=d\beta\otimes s+(-1)^{\deg\beta}\beta\wedge Ds.
$$
The <curvature form of a connection> is $\Theta_D=D^2$. Applying this twice to $fs$ shows that the $df$ terms cancel, so $D^2$ is tensorial and defines an $\operatorname{End}(E)$-valued two-form. With column coordinates for sections and <connection one-form> $A$, the local expression is $D=d+A\wedge$. Direct expansion on an arbitrary bundle-valued form gives
$$
(d+A\wedge)^2=dA\wedge+A\wedge A\wedge.
$$
The terms containing a derivative of the argument cancel by the graded <Leibniz rule>. Thus <Cartan curvature matrix equation> is
$$
\boxed{\Theta=dA+A\wedge A.}
$$
Here multiplication includes matrix multiplication and the <exterior product> of form entries. For a frame change $e'=eg$, the <connection one-form> and <curvature form of a connection> transform as
$$
A'=g^{-1}Ag+g^{-1}dg,\qquad
\Theta'=g^{-1}\Theta g.
$$
The <trace> is consequently frame independent. Also
$$
\operatorname{Tr}(A\wedge A)=\sum_{i,j}A_{ij}\wedge A_{ji}=0:
$$
the diagonal terms vanish and the off-diagonal terms cancel in pairs. Locally $\operatorname{Tr}\Theta=d\operatorname{Tr}A$, and hence $d\operatorname{Tr}\Theta=0$. The local primitives need not agree, but the two-form does. We obtain \b[a global closed two-form] $\operatorname{Tr}\Theta_D$.
The <determinant connection> on $\det E=\bigwedge^rE$ is defined intrinsically by
$$
D^{(r)}(s_1\wedge\cdots\wedge s_r)
=\sum_{j=1}^r s_1\wedge\cdots\wedge Ds_j\wedge\cdots\wedge s_r,
$$
where the differential-form coefficient of $Ds_j$ is placed first. In a local frame, only the diagonal components contribute to the derivative of $e_1\wedge\cdots\wedge e_r$, so its <connection one-form> is $\operatorname{Tr}A$. A line-bundle connection has curvature $d\operatorname{Tr}A$, because a scalar one-form wedges with itself to zero. Therefore
$$
\boxed{\Theta_{D^{(r)}}=\operatorname{Tr}\Theta_D.}
$$
To see independence of the <de Rham cohomology> class, write $D_1-D_0=a$, a globally defined endomorphism-valued one-form. The <curvature difference formula> and the same trace cancellations give the <trace curvature transgression>
$$
\operatorname{Tr}\Theta_{D_1}-\operatorname{Tr}\Theta_{D_0}
=d\operatorname{Tr}a.
$$
Their difference is globally an <exact differential form>, so their <de Rham cohomology> classes coincide.
For the <Hermitian metric on a holomorphic vector bundle>, use the displayed ordering of the local matrices and put $B=(\partial h)h^{-1}$. The identity $\bar\partial h^{-1}=-h^{-1}(\bar\partial h)h^{-1}$ gives
$$
\bar\partial B
=(\bar\partial\partial h)h^{-1}
+(\partial h)h^{-1}\wedge(\bar\partial h)h^{-1}.
$$
The plus sign comes from applying the graded <Leibniz rule> to the one-form $\partial h$. By the derivative formula for a <determinant>,
$$
\operatorname{Tr}B=\partial\log\det h.
$$
The <Hermitian positive-definite matrix> $h$ has positive real determinant, so this logarithm is an ordinary smooth real function. The <trace of Chern curvature> in these conventions is therefore
$$
\boxed{\alpha=\bar\partial\partial\log\det h
=d(\partial\log\det h)=-\partial\bar\partial\log\det h.}
$$
It is locally an <exact differential form> and in particular closed.
A <holomorphic local frame> change multiplies $\det h$ by $|\det g|^2$. A nowhere-zero <holomorphic function> has a local <holomorphic logarithm>, so $\bar\partial\partial\log|\det g|^2=0$. Thus the preceding expression for $\alpha$ is independent of the frame and glues globally. For two <Hermitian metrics on a holomorphic vector bundle>, the function
$$
f=\log\frac{\det h_1}{\det h_0}
$$
is globally defined because the frame-change factors cancel. Their two-forms differ by $d(\partial f)$. Hence \b[the class of $\alpha$ is independent of the metric]. The two-form is generally imaginary-valued; the metric independence statement is in complex <de Rham cohomology>, or equivalently for the real form $i\alpha$.
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