It is difficult to add content to a page using a phone when the page is very large. Fragments of content will be built here and then transported over later, when working on the laptop. The first phase of Chasing Rabbits didn’t any particular end goal. That changed when a statement was found stating that a transformation matrix wasn’t a tensor. That led to the question “well, can we put some time into studying mathematical objects that actually are tensors?” That was the start of Phase Two, and it led to the Kronecker Delta and the Levi-Civita Symbol.
Feed your head!
Tensor tutorials almost immediately ask students to abandon matrix notation and start using index notation. This change is important, for matrix notation is cumbersome. However, students at Mockingbird Academy are exploring the idea that maybe we should show a few more things with matrix notation.
The matrix below is intended to show that transformation could be as simple as changing inches to centimeters.
This should be a surprise. We might judge tutorials we read in the future, harshly, for not mentioning this. We might also feel some regret that the literature that let this cat out of the bag didn’t follow immediately with an explanation what it would take for something that looks like it could be a tensor–to be an actual tensor.
Somehow, we know that hunting for such an explanation is going mean we’re going hunting for more rabbits.
Rabbit Pages
A matrix L operates on old basis vectors to make new basis vectors. The transpose of L will operate on old Vector coordinates to make new vector coordinates.
Linearity of outer product calculations.
Transformation matrix using partials for the notation of the components.
We see how we could increase tensor rank and do it, using nesting, only in the up and down direction.
This could give us the option of recognizing that indices can be in one of two categories and using up-and-down for one and left-and-right for the other. This would keep everything on a page of paper.
Work is done to examine the basic meaning of common words such as Algebra or Calculus keeping in mind that there is a lot more to these things than what we learn in a year of high school (for Algebra) or three semesters of college (for Calculus).
in progress
in progress —
We take the unusual step of assigning a number to the units of a measurement…
10 (1 cm)
Over time we started noticing a story where a tool works on something old to make something new.
We write kx = y (instead of y=kx) to show you that k changed x, making it into y.
We might write kx = x’ where ‘ indicates the new one and lack of apostrophe indicates the old one.
We look at a viewpoint of one thing being projected onto another…
Quest(s)
We are hunting to find the page where we learn that in order for what we want to be true a certain matrix must be such that its inverse is also its transpose.
Rabbits Two had that work.
Testing Zone
The original idea for quick testing still applies. Can we write a prime on an index?
We are also looking at Cartesian Tensors for that last content that we need for our tensor story.
The story White Rabbit One does not qualify as a chasing rabbits story, but you may want to read it.
Finally, if you like the Chasing Rabbits aspect of Vectoreverything, you will probably like our web site Alice in Tensorland.
Vectoreverything 