Invariance and Integrity

  • Not every matrix is a rank 2 tensor.
  • Not every vector is a rank 1 tensor.

“I am a rank 2 tensor. That matrix over there is just a collection of numbers.”

If you aspire to be a vector, you go through a process where you say “I want components–what numbers do I need, so that when they are multiplied against my basis vectors, I am the vector I claim to be?”

A vector needs to be scaled to units, and this might be done by drawing the vector on graph paper. If this route is taken, a vectors length can be given in graph paper units and additional information may be available in the form of relating a unit of graph paper to a known unit of length such as inches.

If we want to draw a vector with a length of 10 inches, we will probably draw it to 10 units and in some way tether the information that 1 unit corresponds to 1 inch.

When we are taught about one-forms, we can look at an example of the math (we show one below) and see that the one form is scaled to the unit vector. In the problem below \begin{bmatrix} 5 \\ 0 \end{bmatrix} is the vector and \begin{bmatrix} 2 \dfrac {x} {\sqrt{x^2 + y^2}} && 2 \dfrac {y} {\sqrt{x^2 + y^2}} \end{bmatrix} is the one-form.

\begin{bmatrix} 2 \dfrac {x} {\sqrt{x^2 + y^2}} && 2 \dfrac {y} {\sqrt{x^2 + y^2}} \end{bmatrix} \begin{bmatrix} 5 \\ 0 \end{bmatrix} = \begin{bmatrix} 10 \\ 0 \end{bmatrix}

representation of vector (5,0) and one-form (2,2)

The above one-form is (2,0).