Two-form

A two-form is a bilinear form.

One form content

Cartoon: A one-form, \tilde w , eats a vector \overrightarrow v and turns into a scalar.

We would like to introduce an example so you can already have it in mind when you read the explanation of a one form.

Assume you are driving down the highway at 60 miles per hour and this highway has mile markers and you started a time equals zero. Every minute you drive past a mile marker.

One day they need to do severe road construction on the entire highway so they put four signs per mile equally spaced and each sign reads “Caution: Drive Slowly, 15 mph.”

60/4=15

60/15=4

Notice how at 15 mph you will drive past a caution sign every minute?

Notice how at this point the store we have two different metrics for length. We still have the mile markers, and we have to caution flag markers.

We can argue that the number four comes from doing math involving the link between mile markers mile markers and the length between caution Flags.

Now, let us assume vector length corresponds to the distance we want to travel. Let us decide that we want the scalar to be time in minutes and therefore our one-form is constrained to be whatever is necessary to make this possible.

We know enough at this point to say that the action of the one-form is mathematically taking the magnitude of the vector and dividing that by four.

Assume our vector space is comprised of the the column vectors fitting the following template:

\overrightarrow v = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}

The dual vector space will be one-forms that are row vectors of the form below:

\tilde w = \begin{bmatrix} b_1 && b_2 && b_3 \end{bmatrix}\begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}

\tilde w ( \rightarrowover v l) = \begin{bmatrix} b_1 && b_2 && b_3 \end{bmatrix}

We get a scalar when we multiply a row matrix and a column matrix.