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Assume $M$ is a smooth manifold, $g:[0,1]\to M$ is a smooth curve on $M$, and $w$ is $1$-form on $M$.

Definition: $$\int_gw=\lim_{\Delta\to 0}\sum_{i=1}^nw(x_i)$$

The tangent vectors $x_i$ are constructed in the following way. The interval $[0,1]$ is partitioned by the points $t_i$. The interval $\Delta_i=[t_i,t_{i+1}]$ we can think of as a tangent vector to $[0,1]$ at the point $t_i$. Its image in the tangent space to $M$ at $g(t_i)$ is $x_i:=\text{d}g|_{t_i}(\Delta_i)$.

Questions: How do we look at $\Delta_i$ as a tangent vector? Is $\Delta_i$ the tangent vector $(t_{i+1}-t_i,t_i)$? How do we compute $x_i$, (assuming tangent vectors are derivations of the algebra of smooth functions, or velocities of curves)?

Why does the limit always exist as the diameter $\Delta$ of the partition goes to $0$?

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I guess you forgot the $dg_{t_i}(\Delta_i)$ in the definition, which approximates the tangent vector –  Giuseppe Negro May 6 at 18:18

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Imagine for a moment that the manifold $M$ is embedded in some flat, ambient vector space $V$. The points $t_i$ are therefore vectors in $V$, and the difference between adjacent points is also a vector in $V$. Moreover, as $\Delta \to 0$, the difference vector $t_{i+1} - t_i$ becomes more and more intrinsic to the manifold $M$--more and more like a tangent vector, which it becomes in the limit.

Without invoking an embedding, though, I would've just taken the derivative of $g$. $g'$ is a velocity, and so it's guaranteed to be a tangent vector everywhere on the smooth, differentiable curve. Taking such a position, you get this:

$$\int_g w = \int_0^1 w(g'(t)) \, dt$$

One should take care to understand that the $dt$ here merely tells you integrate with respect to $t$; it isn't a basis form on $M$. Writing the integral this way allows you to see that the integral of a 1-form on a curve can be understood as follows: the 1-form evaluated at each point eats the tangent vector at each point, and what we integrate is that scalar function using the usual tools of multivariable calculus.

This idea generalizes to $k$-forms integrated on $k$-dimensional oriented objects: curves, surfaces, volumes, and so on.

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