the Hookah-smoking Caterpillar talks

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Introduction

Not just anyone can fall down the rabbit hole.  You have to be like Alice.  You have to want to find the weird stuff and to see it as an opportunity to try and make sense out of all the "crazy".  Welcome!

Restriction to Addition of Scaled Quantifiables

A math person would protest and say that something like "3" or "4" is a  quantity.  After a few puffs of hookah we will tell you that the "3" or "4" has to be scaling something that is quantifiable, something for which there exists a quantity given the number "1"...

...like a small chocolate eclair...

If we find three of them here, and four of them there, then we can put them all together and  we can do a legitimate addition:

3 eclairs + 4 eclairs = 7 eclairs

If we look at addition this way then what we learn about vectors in Linear Algebra will be more sensible.  We will look at something like (3,4) and think that the 3 and the 4 must be scaling things--that there is more to it than just the numbers.

To make life less tedius, if you write 3+4=7 we will assume that it stands for 3 units + 4 units = 7 units.  You will not be scolded or punished.

Conjugacy of Vectors and One-forms

The idea of one-forms is a real trip (see also co-vectors).  A one-form is a vector and yet it is different enough from a vector that we say the interaction of a vector with a one-form makes a scalar.  We don't get a scalar from doing the same math on two vectors (or from two one-forms).

Part of this magic appears when we write

ax + by = k

If (x,y) is the vector then (a,b) is a one-form.  If we tell this story and we are in the vector space and over in the dual vector space the caterpillar is telling everyone that (a,b) is the vector and that (x,y) is the one-form.

Further reading (a Cal Tech PDF)

Representations

There is magic in the noun representation.  Representation is the "trick" of doing work with something that is simpler than the original.  We can do math work to get the answer and then convert what we get back to the original.  You get to choose how you use the time that you saved.

As a quick example, for vectors, the original might be ...

3(^i) + 4(^j)

...and we choose to represent it as (3,4).  Later when we need to do an addition we just add the representations...

(3,4) + (1,2) = (4,6)

...and for our last step we convert (4,6) to...

4(^i) + 6(^j)

Representation is a rabbit hole and we have farther to fall...

Assume that we represent a vector with its components (3,4) and we also represent a one form with its components (1,2).  Remember the (^i) and (^j) that we had for the vector earlier?  We have similar "math things" for the one-form.

We are invoking Representation if we decide to represent (3,4) by matrix and to represent (1,2) by another matrix.  So, yes, we are taking a representation of a representation.  Sounds crazy, right?

The sanity is that we end up doing a matrix multiplication (a row and then a column) and you know from Linear Algebra that this gives us a scalar.

The work you do where Matrix Multiplication mimics Tensor Calculus is can be thought of as the time you spend learning to ride a bike by using a bike that has training wheels on it.  It's helpful but in a way you hate it and can't wait to get rid of it.

Matrix multiplication depends on putting things together in the correct order (the row before the column).    Tensor Calculus doesn't need this.  If we are jumping back and forth between tensor calculus and matrix multiplication then we had build in a restriction so that we put the tensors in the correct order before we do the the math.

We don't need the positioning in Tensor Calculus because every component in each tensor knows where it needs to go in the calculation.  This is especially crazy and we believe it is best seen by programming a computer to do the math and seeing the FOR loops of the computer program.  The computer produces the correct answer and the computer can't "see" your rows and columns--it relies on every component having index numbers.