# Concrete example for one forms.

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.

• You might find my YouTube lectures helpful. Note that there is a complete version of the first differential forms lecture at the very bottom of the long list. Dec 9 '19 at 20:27

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*}

• Thanks for that really quick and great answer. Since I can't seem to find an example that basic, could you also tell me how I would wedge the two $\omega_1$ and $\omega_2$ from above?
– MJP
Dec 9 '19 at 22:04
• Sure thing: note that $\,dx \wedge \,dx = 0$. Then with $\omega_1 = \,dx + 2 \,dy - 4 \,dz$ and $\omega_2 = x \,dx + 2y^2 \,dy - 3xz \,dz$, we have: $\omega_1 \wedge \omega_2 = x \,dx \wedge \,dx + 2x \,dy \wedge \,dx - 4x \,dz \wedge \,dx + 2y^2 \,dx \wedge \,dy + 4y^2 \,dy \wedge \,dy - 8y^2 \,dz \wedge \,dy - 3xz \,dx \wedge \,dz - 6xz \,dy \wedge \,dz + 12xz \,dz \wedge \,dz$. Dec 9 '19 at 22:15
• Thanks, great help.
– MJP
Dec 9 '19 at 22:17