# Do functions distribute over the wedge product?

I am wrapping my head around the arithmetic properties of the wedge product. I understand that constants do distribute over the wedge product, i.e. for $$c_1,c_2\in\mathbb{R}$$, it holds $$$$c_1 \omega_1 ∧ c_2 \omega_2 = (c_1\cdot c_2) \omega_1 ∧ \omega_2$$$$ Does this still apply, when the coefficients are functions? Does the following equation hold? $$$$(\cos(x) dx) ∧ (y dz) = (\cos(x)\cdot y) dx∧dz$$$$ Thank you very much in advance!

They do. If you have the wedge product as an assignment of some member of the (nth) exterior product over the cotangent bundle to each point on your space ($$\mathbb{R}^n$$ or whatever it may be), then something like $$xdy\wedge ydz$$ is just a representation of this assignment which just so happens can be represented by the exactly same expression everywhere. The form is just a fixed function at any given point in the space, $$x,y,z$$ etc are not changing and so act exactly the same as constants in the written expression.
It's worth remembering that functions are 0-forms. So when we write $$f\omega$$, we're really writing $$f \wedge \omega$$, where $$f$$ is a 0-form and $$\omega$$ is a $$k$$-form, and the product is a (new) $$k$$-form. Now let's look at $$\alpha = (f \omega) \wedge (g \eta)$$ where $$\eta$$ is a $$p$$-form, and $$g$$ is again a real-valued function. Writing this out with wedges \begin{align} \alpha &= (f \wedge \omega) \wedge (g \wedge \eta)\\ &= f \wedge \omega \wedge g \wedge \eta & \text {because wedge is associative}\\ &= f \wedge g \wedge \omega \wedge \eta & \text{applying product rule to \omega \wedge g}\\ &= (fg) (\omega \wedge \eta) \end{align}
The product rule says that $$\phi \wedge \tau = (-1)^s \tau \wedge \phi$$, where remembering the rule for $$s$$ is the tricky bit. It turns out to be the product of the degree of $$\tau$$ and the degree of $$\phi$$. In the case where $$\phi$$ is a $$0$$-form, one of these degrees is zero, so $$s = 0$$, and so $$0$$-forms commute with other forms without any sign-change. Yay!