[proofplan] We prove both implications by translating orientations into choices of ordered local frames. If an atlas has positive transition determinants, the standard orientation of $\mathbb R^k$ can be transported to each fibre, and positivity guarantees that this transported orientation is independent of the chosen trivialization. Conversely, an orientation on $E$ allows us to choose local frames that are positively oriented in every fibre; the transition matrix between two positively oriented frames has positive determinant, giving the desired equivalent atlas. [/proofplan]

[step:Transport the standard orientation through an atlas with positive transition determinants]Assume first that there is a bundle atlas $\{(V_a,\Psi_a)\}_{a\in A}$ with transition maps $h_{ab}:V_a\cap V_b\to GL(k,\mathbb R)$ satisfying $\det h_{ab}(x)>0$ for every $x\in V_a\cap V_b$. For each $a\in A$ and each $x\in V_a$, define an ordered basis \begin{align*} e_{a,1}(x),\dots,e_{a,k}(x)\in E_x \end{align*} by \begin{align*} \Psi_a(e_{a,r}(x))=(x,\varepsilon_r) \end{align*} for $1\le r\le k$, where $\varepsilon_1,\dots,\varepsilon_k$ is the standard ordered basis of $\mathbb R^k$. We declare an ordered basis of $E_x$ to be positive if it has the same orientation as $e_{a,1}(x),\dots,e_{a,k}(x)$ for one, hence every, $a$ with $x\in V_a$. To check independence of $a$, let $x\in V_a\cap V_b$. By definition of the transition map, \begin{align*} \Psi_a\circ \Psi_b^{-1}(x,v)=(x,h_{ab}(x)v). \end{align*} Thus the change-of-basis matrix from the frame $e_{b,1}(x),\dots,e_{b,k}(x)$ to the frame $e_{a,1}(x),\dots,e_{a,k}(x)$ is $h_{ab}(x)$. Since $\det h_{ab}(x)>0$, these two ordered bases determine the same orientation of $E_x$. Therefore the above rule gives a well-defined orientation on every fibre. The local frames $e_{a,1},\dots,e_{a,k}:V_a\to E$ vary smoothly because they are obtained from the smooth trivialization $\Psi_a^{-1}$ applied to constant coordinate vectors. Hence the fibre orientations are locally represented by smooth positive frames. This is precisely an orientation of the real vector bundle $E$.[/step]

[guided]Assume that the bundle has an atlas $\{(V_a,\Psi_a)\}_{a\in A}$ whose transition maps have positive determinant. The standard ordered basis of $\mathbb R^k$ is denoted \begin{align*} \varepsilon_1,\dots,\varepsilon_k\in \mathbb R^k. \end{align*} For each chart index $a\in A$ and each point $x\in V_a$, the trivialization $\Psi_a:\pi^{-1}(V_a)\to V_a\times\mathbb R^k$ gives an ordered basis of the fibre $E_x$ by pulling back the standard basis: \begin{align*} \Psi_a(e_{a,r}(x))=(x,\varepsilon_r), \qquad 1\le r\le k. \end{align*} The intended orientation on $E_x$ is the one for which this ordered basis is positive. The only point requiring verification is that this definition does not depend on which trivialization contains $x$. Suppose $x\in V_a\cap V_b$. The transition map $h_{ab}:V_a\cap V_b\to GL(k,\mathbb R)$ is defined by \begin{align*} \Psi_a\circ\Psi_b^{-1}(x,v)=(x,h_{ab}(x)v). \end{align*} This formula says exactly that the coordinate vector of an element of $E_x$ in the $a$-frame is obtained from its coordinate vector in the $b$-frame by multiplying by $h_{ab}(x)$. Therefore $h_{ab}(x)$ is the change-of-basis matrix from \begin{align*} e_{b,1}(x),\dots,e_{b,k}(x) \end{align*} to \begin{align*} e_{a,1}(x),\dots,e_{a,k}(x). \end{align*} Two ordered bases of a real $k$-dimensional [vector space](/page/Vector%20Space) determine the same orientation exactly when the determinant of their change-of-basis matrix is positive. Since $\det h_{ab}(x)>0$, the two local rules give the same orientation on $E_x$. Thus the fibre orientation is well defined. Because each frame $e_{a,1},\dots,e_{a,k}$ is obtained by applying $\Psi_a^{-1}$ to the constant standard frame on $V_a\times\mathbb R^k$, it varies smoothly on $V_a$. Hence these locally smooth positive frames define an orientation of $E$.[/guided]

[step:Choose positively oriented local frames from an orientation] Conversely, assume that $E$ is oriented. By the definition of an orientation of a real vector bundle, for every point $p\in M$ there is an open neighbourhood $V_p\subset M$ and a smooth local frame \begin{align*} s_{p,1},\dots,s_{p,k}:V_p\to E \end{align*} such that, for every $x\in V_p$, the ordered basis \begin{align*} s_{p,1}(x),\dots,s_{p,k}(x) \end{align*} is positive for the chosen orientation of $E_x$. Each such frame defines a local trivialization \begin{align*} \Psi_p:\pi^{-1}(V_p)&\to V_p\times\mathbb R^k \end{align*} by requiring \begin{align*} \Psi_p\left(\sum_{r=1}^k v_r s_{p,r}(x)\right)=(x,(v_1,\dots,v_k)) \end{align*} for every $x\in V_p$ and every $(v_1,\dots,v_k)\in\mathbb R^k$. These trivializations form a bundle atlas for $E$, and it is equivalent to the original atlas because all local trivializations of the same smooth vector bundle have smooth $GL(k,\mathbb R)$-valued transition maps on overlaps. [/step]

[step:Show that transitions between positive frames have positive determinant] Let $p,q\in M$, and let $x\in V_p\cap V_q$. Let \begin{align*} h_{pq}:V_p\cap V_q\to GL(k,\mathbb R) \end{align*} be the transition map determined by \begin{align*} \Psi_p\circ\Psi_q^{-1}(x,v)=(x,h_{pq}(x)v). \end{align*} Equivalently, the matrix $h_{pq}(x)$ is the change-of-basis matrix from the ordered basis \begin{align*} s_{q,1}(x),\dots,s_{q,k}(x) \end{align*} to the ordered basis \begin{align*} s_{p,1}(x),\dots,s_{p,k}(x). \end{align*} Both ordered bases are positive for the given orientation of $E_x$. Therefore their change-of-basis determinant is positive: \begin{align*} \det h_{pq}(x)>0. \end{align*} Since $p,q$ and $x\in V_p\cap V_q$ were arbitrary, every transition function in the atlas $\{(V_p,\Psi_p)\}_{p\in M}$ has positive determinant. Thus $E$ admits an equivalent atlas with positive transition determinants. [/step]

[step:Conclude the equivalence] The first implication constructs an orientation from any atlas whose transition determinants are everywhere positive. The second implication constructs such an atlas from any given orientation by choosing positively oriented local frames. Hence $E$ is orientable if and only if it admits an equivalent bundle atlas whose transition functions have positive determinant everywhere on overlaps. [/step]