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I'm learning differential forms (via the excellent book 'A Visual Introduction to Differential Forms and Calculus on Manifolds' by J. P. Fortney).

One operation I'm not 100% I understand how to parse is the notion of multiplying a form, say a 2-form, by a function e.g as in Question 3.31 in the book:

enter image description here

I get that that by using the definition in the book as a starting point, namely that the value of the two form dz^dx(u, v) itself can be computed essentially as a determinant (of the relevant projected components of the two input vectors), I get a real number representing the oriented volume of the projected parallelogram on the Z-X plane.

However, once I add in the 'xyz' bit infront, how do I treat this object? are those x, y, z components referring to the point 'p' at which I'm working with the cotangent space? e.g. so the whole thing is a function of the point i'm tangent at (plus the two vectors given to the two form as input, comprising in this case essentially a constant vector field)?

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    $\begingroup$ The multiplication is pointwise. You evaluate the form on some tangent vectors at a point to get a number, then evaluate the function at that same point to get a number, then multiply these numbers. $\endgroup$ Oct 26, 2022 at 8:49

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Same thing as Qiaochu said I'm just adding a bit of details for completeness.

If $\omega$ is a differential form on your manifold $M$, then at any point $p \in M$, it is essentially a map:

\begin{equation} \omega|_p : T_p M \times \dots \times T_p M \to \mathbb{R} \end{equation} Given a function $f : M \to \mathbb{R}$, then the form $f \omega$ is defined pointwise:

\begin{equation} (f\omega)|_p = f(p)\cdot\omega|_p \end{equation}

where $\cdot$ just denotes usual multiplication in $\mathbb{R}$ (since $\omega|_p$ outputs a real number).

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  • $\begingroup$ Got it, yes that’s what I surmised. Thank you for clarifying! $\endgroup$
    – giorgio
    Oct 26, 2022 at 10:02

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