I learned about orientations on manifolds and induced orientation on the surface and I don't get the idea. When I search online I find explanations with different definitions.
Here is how we defined orientaion:
The induced orientation was defined using compostion of the orientation with the containment function multiplied with $(-1)^m$ where m is the dimension of the manifold.
What is the $(-1)^m$ for? I would be glad to get an intuitive explanation and an example of induced orientation on a simple manifold (preferably one from odd dimension).