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So integration of a 1-form $\omega$ over a path $\gamma$ is defined to be the integral of the pullback of $\omega$. Why does this make sense? Why don't we integrate over a vector field instead, like in vector calculus, and define integration of a vector field to be the usual integral over its pullback?

Also, a reference for the intuition behind differential forms would be nice.

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6  
Vector fields don't pull back. In vector calculus a lot of things are identified which shouldn't be because you often implicitly work with an inner product. –  Qiaochu Yuan Aug 18 '11 at 3:36
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I remember liking this exposition of Terry Tao's. –  Dylan Moreland Aug 18 '11 at 3:56
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Tao's exposition is quite on-the-mark. Forms are things that are designed to be the minimal thing that you can integrate. Vector fields you can integrate as well, but it requires additional structure to make sense of the integral if you want to use this set-up. Quite often you do have this additional structure so you do integrate vector fields, like in vector calculus. –  Ryan Budney Aug 18 '11 at 4:28
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