The definition of integral of a $k$-form $\omega$ over a parametrized manifold $ Y_\alpha $ is $ \int_{Y_\alpha}\omega = \int_A\alpha^*\omega$ where $ \alpha\colon A \to R^n $.
Let $ \gamma:(a, b) \to R^3$. $ C_\gamma $ is a curve in $ R^3 $ and the line integral over that curve is written as $ \int_{C_\gamma}Pdx + Qdy + Rdz$ where $P$, $Q$ and $R$ are component function of $\gamma $. By applying the definition of integration of a $k$-form over a manifold we should get the formula $ \int_a^bP(\gamma(t)) \frac{d\gamma_1}{dt} + Q(\gamma(t)) \frac{d\gamma_2}{dt} + R(\gamma(t)) \frac{d\gamma_3}{dt}dt.$ However, I seem to be missing something. Here's my computation: $ \int_{C_\gamma}Pdx + Qdy + Rdz = \int_a^bP(\gamma(t))dx(\gamma(t))(\alpha_*(t; v)) + ... \\= \int_a^bP(\gamma(t))dx(\gamma(t))(\gamma(t);D\gamma(t) v) + ... $
where $v$ is some vector, in this case just a scalar, since it has only one component. Obviously, the disappearance of $v$ would give the expected result, but I can't see why it should disappear. I know I'm missing something quite simple, but could someone point that out for me, cause I'm feeling quite helpless right now...
For reference: I'm using the book "Analysis on manifolds" by Munkres.