Concrete example for one forms.

Question

I'm reading about k-forms right now but am still quite confused about the different notations, which is why I'm trying to look at a concrete example. Let's say we have a one-form in three-dimensions: $\omega(x) = dx + 2dy - 4dz$, and for example the vector $\vec{a} = (1,1,1)$, how would I compute $\omega(\vec{a})$? Is that a reasonable question to ask? Sorry if it's confusing, but I'm really just trying to make sense of this by self study which is quite hard.

Answer 1

If your form is $\omega = \,dx + 2 \,dy - 4 \,dz$ and your tangent vector at the point $a$ is $t_a = (1,1,1)$, then you can think about each of the $\,dx, \,dy, \,dz$ as projecting onto the corresponding components of the tangent vector.

In your example, \begin{align*} \omega(t_a) &= \,dx(1,1,1) + 2 \cdot \,dy (1,1,1) - 4 \,dz(1,1,1) \\ &= 1 + 2 \cdot 1 - 4 \cdot 1 \\ &= -1 \end{align*}

Let's look at another example. If your point is $a = (1,2,3)$ and your tangent vector at $a$ is $t_a = (3,2,1)$ and we define $\omega = x \,dx + 2y^2 \,dy - 3xz \,dz$, then \begin{align*} \omega(t_a) &= 1 \,dx (3,2,1) + 2(2)^2 \,dy(3,2,1) - 3 \cdot 1 \cdot 3 \,dz (3,2,1) \\ &= 1 \cdot 3 + 8 \cdot 2 - 9 \cdot 1 \\ &=10. \end{align*}