TITLE: Defining Jacobian of differential map on Riemannian manifold
QUESTION [5 upvotes]: I'm reading Rufus Bowen's notes on ergodic properties of Anosov diffeomorphisms, and I've come across an important-seeming function but I'm confused about how it's defined. 
Let $M$ be a Riemannian manifold, and let $f : M \to M$ be a $\mathcal{C}^2$ Axiom A diffeomorphism. (For our purposes, this means there is a closed invariant $\Omega_s \subseteq M$ for which there is a $Df$-invariant splitting of the tangent bundle $T\Omega_s = E^s \oplus E^u$.) Bowen writes the following (first paragraph of Section 4B): 

For $x \in \Omega_s$, let $\phi^{(u)}(x) = -\log \lambda(x)$, where $\lambda(x)$ is the Jacobian of the linear map $$Df : E^u_x \to E^u_{fx}$$ using inner products derived from the Riemannian metric.

My question is, how exactly is $\lambda(x)$ defined? My first thought was it's the determinant of the Jacobian matrix of partial derivatives, but (a) this requires writing $Df$ in coordinates and I'm not sure the determinant is coordinate-idnependent, and (b) this determinant has seemingly nothing to do with the Riemannian structure on $M$, but the dependence of $\phi^{(u)}(x)$ on the Riemannian metric plays a key role in subsequent arguments of his lecture notes. 
So how do we typically define the Jacobian of the differential of a diffeomorphism? Any help is appreciated!

REPLY [5 votes]: levap has already given a very nice answer. I will give an alternative discussion based on the determinant as a ratio of volume elements. Hopefully this will be useful for those reading Bowen- the interpretation of determinant as a volume ratio is the reason for its importance in smooth ergodic theory.
Let $T : V \to V'$, where $V, V'$ are inner product spaces. A choice of inner product on $V$ gives rise to a choice of scaling of Lebesgue measure $m$, i.e., the one assigning mass $1$ to an orthogonal $\dim(V)$-cube of side length $1$. Let $m'$ be the corresponding Lebesgue volume for $V'$.
With $m, m'$ as above, the determinant $\det(T)$ is (or, more accurately, may be characterized as) a volume ratio: we have
$$
\det(T) = \frac{m'(T U)}{m(U)}
$$
where $U \subset V$ is any nonempty open subset. 
In this way, we see directly how the inner product structures on each of $V, V'$ affect the corresponding notion of determinant. Indeed, taking a different view, we need not regard $V, V'$ as inner product spaces- for the purposes of having a well-defined determinant, it suffices to specify scalings of Lebesgue measure on each of $V, V'$. 
Returning to Bowen's context: The determinant of $df$ between tangent spaces depends on the volume elements on each of $T_x M, T_{fx}M$- these are precisely the scalings of Lebesgue measure specified by the inner products (a.k.a. Riemannian metrics) on each of $T_x M, T_{f x} M$.