Workshop

Work on something below ran into a question, and it is also ran into concern that it would take too long to finish.

We presently believe that what is below shows how we transform the components of basis vectors–when the basis vectors are transformed, the components of the basis vectors must change. We have discovered that it is complicated to actually show and explain everything, especially considering that the reader has only recently started to think about this topic.

Our plan now is to see if we find other web pages that deal with this specific topic (transformation of the components of basis vectors), and how they did it.

Let’s assume we start off with inches because our company is an old, prestigious engineering firm. Along comes the 1970s and we are encouraged to “Go metric!” we make a transformation matrix to go from inches to centimeters.

Assume that two years later, a big customer asks us to go from centimeters to meters, because if you are working in MKS units (meter, kilogram, second) then you like meters. They just won a big contract to build a new Mars Rover — yes, we want in. Yes, we will make that second transformation matrix!

  • [e_1, e_2, e_3] is in inches
    • These are scaled to the graph paper so the constants are each “1”.
  • [\tilde e_1, \tilde e_2, \tilde e_3] is in centimeters
    • Because the basis vectors got shorter, the basis components compensated by getting larger, and all nonzero components are “2.54”.
  • [\hat e_1, \hat e_2, \hat e_3] is in meters
    • Because the basis vectors got larger by two orders of magnitude, the basis components compensated by getting shorter by two orders of magnitude, so all nonzero components are now “0.0254”.

Let A take us from inches to centimeters

[e_1, e_2, e_3] \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} = [ \tilde e_1, \tilde e_2, \tilde e_3]

[ \tilde e_1, \tilde e_2, \tilde e_3] \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix} = [ \hat e_1, \hat e_2, \hat e_3]

We can combine these as follows:

[e_1, e_2, e_3] \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix} = [ \hat e_1, \hat e_2, \hat e_3]

We rearrange the equation to put the quantity defined on the left:

[ \hat e_1, \hat e_2, \hat e_3] = [e_1, e_2, e_3] \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix}

We will insert letters for indices, one at a time. The math work is done on the right side so we will start there. What goes across ‘a’ must go down ‘b’ because of the way we do matrix multiplication:

[ \hat e_1, \hat e_2, \hat e_3] = [e_1, e_2, e_3] \begin{bmatrix} a_{1m} & a_{1m} & a_{1m} \\ a_{2m} & a_{2m} & a_{2m} \\ a_{3m} & a_{3m} & a_{3m} \end{bmatrix} \begin{bmatrix} b_{m1} & b_{m2} & b_{m3} \\ b_{m1} & b_{m2} & b_{m3} \\ b_{m1} & b_{m2} & b_{m3} \end{bmatrix}

What goes across the basis vectors must go down the A matrix:

[ \hat e_1, \hat e_2, \hat e_3] = [e_n, e_n, e_n] \begin{bmatrix} a_{nm} & a_{nm} & a_{nm} \\ a_{nm} & a_{nm} & a_{nm} \\ a_{nm} & a_{nm} & a_{nm} \end{bmatrix} \begin{bmatrix} b_{m1} & b_{m2} & b_{m3} \\ b_{m1} & b_{m2} & b_{m3} \\ b_{m1} & b_{m2} & b_{m3} \end{bmatrix}

We are done with the indices that will participate in summation. There is one more indexing that we can do, and we’ll give it ‘p’:

[ \hat e_p, \hat e_p, \hat e_p] = [e_n, e_n, e_n] \begin{bmatrix} a_{nm} & a_{nm} & a_{nm} \\ a_{nm} & a_{nm} & a_{nm} \\ a_{nm} & a_{nm} & a_{nm} \end{bmatrix} \begin{bmatrix} b_{mp} & b_{mp} & b_{mp} \\ b_{mp} & b_{mp} & b_{mp} \\ b_{mp} & b_{mp} & b_{mp} \end{bmatrix}

\hat e_p = e_n a_{nm} b_{mp}

Other work….

Work to improve the knowledge of tensors is taking help from the following web pages.