I'm reading Frank W.Warner's "Foundations of Differentiable Manifolds and Lie Groups". In page 139, he defines the differentiable map that preserves orientations:
Let M and N be orientable n-dimensional manifolds, and let $\psi: M\to N$ be a differentiable map. We say that $\psi$ preserves orientations if the induced map $\delta \psi:\Lambda_n^*(N)\to \Lambda_n^*(M)$ maps the component $\Lambda_n^*(N)-O$ determining the orientation on N into the component of $\Lambda_n^*(M)-O$ determining the orientation on M. Equivalently, $\psi$ is orientation-preserving if $d \psi$ sends oriented bases of the tangent spaces to M into oriented bases of the tangent spaces to N.
But I can't see what the map $\delta \psi:\Lambda_n^*(N)\to \Lambda_n^*(M)$ is when $\psi: M\to N$ is only a differentiable map. And I think all the arguments make sense if $\psi: M\to N$ is a diffeomorphism.