# How to compute differential forms multiplied by functions

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:

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)?

• 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. Oct 26, 2022 at 8:49

If $$\omega$$ is a differential form on your manifold $$M$$, then at any point $$p \in M$$, it is essentially a map:
$$$$\omega|_p : T_p M \times \dots \times T_p M \to \mathbb{R}$$$$ Given a function $$f : M \to \mathbb{R}$$, then the form $$f \omega$$ is defined pointwise:
$$$$(f\omega)|_p = f(p)\cdot\omega|_p$$$$
where $$\cdot$$ just denotes usual multiplication in $$\mathbb{R}$$ (since $$\omega|_p$$ outputs a real number).