In my ignorance, when first learning, I just assumed that one pushed a vector forward to where a form lived and then they ate each other. And I assumed one pulled a form back to where a vector lived (for the same reason). But I see now this is idiotic: for one does the pullback and pushforward between two DIFFERENT manifolds. But forms and vectors live on the cotangent and tangent space of the SAME manifold. So, that being the case... WHY does one do this? I can sort of intuit why ones does this for functions (pulling back a function to a simpler place). But for forms? Why does one WANT to pull back a form? Why does one WANT to push a vector forward? Examples in dynamcis are most welcome (if possible).