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I am self-learning differential geometry, may I ask the following questions:

  1. If a 1-form is a linear function $\omega :T_p\mathbb{R}^n\to \mathbb{R}$, does it mean it's just dot product of the tangent vector $\left \langle dx_1,...,dx_n \right \rangle$ to some line?

  2. An n-form takes in n vectors and gives a number proportional to the signed volume of the n-dimensional parallelotope. People say they provide something to be integrated with, what does that mean? Do they mean sth like $\text{volume}=\int dx_1 \wedge...\wedge dx_n$, where $\int$ is Lebesgue integral and $dx_1 \wedge...\wedge dx_n$ is n-form?

I hope my questions aren't too stupid. I'm just a beginner. Many thanks in advance!

HIH
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  • $\langle dx_1,\dots,dx_n\rangle$ is not a tangent vector. The partial derivatives $\partial_{x_i}$ (at $p$) are a basis of the tangent space $T_p\mathbb R^n$ and the one-forms $dx_i$ are the basis of the dual space $T_p^*\mathbb R^n,.$
    • – Kurt G. Nov 19 '23 at 15:22
  • Use MSE's search functionality: https://math.stackexchange.com/search?q=lebesgue+differential+form
    • – Kurt G. Nov 19 '23 at 15:22