Hookah Smoking Caterpillar

The Screwdriver vs. the Screw

If this was a shop class you could look at a screwdriver and look at a screw and say “yes, I see the difference.”

We aren’t so lucky here. When you start learning this stuff the text probably starts with talking about tensors and then it moves on to talking about the tools that used to make changes to tensors (changes like going from 10 inches to 25.4 centimeters so the change isn’t really a change).

To the new person, it might seem like everything is a tensor. We apologize for that.

Tensors

Let’s start with the question “How do we know a tensor, is a tensor?”

A tensor might be a tensor because it passes a series of tests.

Assume we want to prove that an integer is even. We can do it with one test: take your integer and divide it by 2. If the answer is an integer, then your integer is even. An integer like 3, which is not even, will give an answer with a fraction (1.5).

A tensor might be a tensor because we created it by starting with a known tensor and we do things to it following a set of rules that restrict what we can do.

To give you an idea of this, suppose you start with an integer and then you are told “you can take your number and add 1 to it”. You can do that as many times as you like, but that is the only thing you can do. When I come back an hour later, you will still have an integer.

Go head and take your first puff of hookah.

The Vector that Isn’t

There are two concerns

  • (3,4)  – This might be a point, it might be just two numbers
  • (3,4) might be certified as something we can use, but we say it is just a representation of a vector (or a tensor)

In the morning I drink a cup of coffee and I like it. That cup of coffee is real. Anything like a photo, or a painting, or a drawing or the words “cup of coffee” printed on paper is just a representation.

In a nightmare, a bull is charging toward you. When it reaches you, you will wake up. It doesn’t matter if a first person measures the distance-to-bull as 0.5 kilometers and a second person measures it as 500 meters and it doesn’t matter if the first person says you have 30 seconds left and the second person says you have 0.5 minutes.

Hopefully you can see that the changes made from first person to second person don’t change the story.

I want you to believe that the story that is real is important. Everything you learn here will be tricks and techniques for working with representations. I want you to have skills for this, both algebraic and geometric. When you see something the word ‘invariant’ in it, I want you to imagine you hear me saying “the story is what is important–the story must not change”.

Take another puff of hookah.

Relative Motion

Your nose has an itch. If you rub it, the suffering will go away for a few minutes.

There are two ways to cure the itch:

  1. Rub the upper part of your nose with your finger.
  2. Rub your finger with the upper part of your nose.

Strictly from the math, both things are the same–and both fix the itch, but if we have a frame of reference we can use the frame of reference to prove that the nose didn’t move or alternately that the finger didn’t move.

Have another puff of hookah.

Construction Zone

You know that the speed on a slower highway id 60 MPH. The distance between highway markers is one minute and the time between adjacent markers is one minute.

The decide to do construction on one part of it and the speed through the construction zone is 15 MPH. They put construction signs every 0.25 miles. Every minute you drive past a construction sign.

The time between signs was preserved by making the distance between signs follow change to the speed.

One more thing to think about–originally there was 1 sign per mile. After construction began there were 4 signs per mile.

When you get to the point where you learn about one forms (they might say co-vectors), think about the story above.

Have another smoke of hookah.

Have another puff of hookah.

Transformations

A vector has a magnitude and it can be thought of as the sum of the changes to the components. In three dimensions that would be changes in the directions of x, y and z.

New Point: \begin{bmatrix} 1 \\ 10\end{bmatrix}

Old Point: \begin{bmatrix} 2 \\ 20 \end{bmatrix}

Vector \begin{bmatrix} 1 \\ 10 \end{bmatrix}

Now assume I tell you the unit vectors that we use for this vector have a length of 1 inch.

Later, someone says they want to change the lengths of the unit vectors to 1 cm.

You think about it and you say “every component will be multiplied by 2.54”.

  • (2.54) 1 = 2.54
  • (2.54) 10 = 25.4

I want you to see what the math looks like for what you just did:

\dfrac { \partial x' } { \partial x} dx = dx'

\dfrac { \partial y' } { \partial y} dy = dy'

The notation \dfrac { \partial x' } { \partial x} is a placeholder for whatever math is needed to take us from x to x’ and in this case from inches to centimeters.

It is legal for the transformation from old coordinates to new coordinates to be “multiplication by a constant”. Don’t forget, the component increased by 2.54 and the unit vector shrank by 2.54–this was true for both x and y.

Transformations are allowed to be more complicated than that. It is allowed to have math where a new coordinate depends on each of the old coordinates (and likewise, a new component depends on each of the old components).

\dfrac { \partial x' } { \partial x} dx + \dfrac { \partial x' } { \partial y} dy + \dfrac { \partial x' } { \partial z} dz = dx'

\dfrac { \partial y' } { \partial x} dx + \dfrac { \partial y' } { \partial y} dy + \dfrac { \partial y' } { \partial z} dz = dy'

\dfrac { \partial z' } { \partial x} dx + \dfrac { \partial z' } { \partial y} dy + \dfrac { \partial z' } { \partial z} dz = dz'

You can think of the above as telling us what we are allowed to do in a transformation. It also tells us what we can’t do. For example, we cannot have something like the following:

dx’ = a(dx)(dx) + b (dx)(dy) + c (dy)(dy)

Something new can’t depend on the square of something old and it cannot depend on the product of two old things.

You have to dive deep into the math to get to the point where you can explain we would want these restrictions. When you read about math, perk your ears up if you see an adjective like “linear”, “orthogonal” or “orthonormal”. If the article says that something is true only if the transformation is linear, whatever it is, is important to you.

You might be thinking that in physics we CAN have something depend on a square. For example, if the acceleration provided by gravity is constant and we start with a zero velocity then the change to position will depend on the square of time.

The deal here is, here we are talking about transformation matrices, the tools that we use that don’t interfere with the story. In an incredibly small amount of time some particle travelled 10 inches and then we are told the customer needs the answer in centimeters so we did the transformation and got 25.4 centimeters.

Congrats that you made it all the way to here — go ahead and smoke the whole pack.

Magic of Matrices

We say that matrices are not rank two tensors. There’s more than one reason to say this.

We started by telling you that if you see brackets with numbers written in them, it might be an imposter–it’s just a bunch of numbers written inside brackets. For something to be a rank 2 tensor it needs to have some geometry.

There’s another consideration. We might put forth the numbers for two matrices, as is done below…

\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix} = ?

(hang on to something and don’t panic)

Different tensors can be represented by the same matrix. The nature of the two tensors represented by the two matrices above will dictate what math we do.

  1. One type of math will return a single number (that is 1 component).
  2. One type of math will return a 2×2 matrix and it has 4 components.
  3. One type of math will deliver a 2x2x2x2 beast that has 16 components.

Time for damage control. Assume we call the first matrix A and we call the second matrix B and we saw whatever the answer is, that is C. The three scenarios mentioned above correspond to the following three notations:

  1. A_{ij} B_{ij} = C
  2. A_{ij} B_{jk} = C_{ik}
  3. A_{ij} B_{kl} = C_{ijkl}

At the very least, you will notice that the subscripts are different each time. That goes along with the idea that we got different math from different tensors.

You might notice that ijkl goes with 2x2x2x2 and ik goes with 2×2

You mind might on impulse start considering the following game:

  • ijkl — 2x2x2x2 = 16 components
    • divide by 2
  • ijk — 2x2x2 = 8 components
    • divide by 2
  • ij — 2×2 = 4 components
    • divide by 2
  • i — 2 = 2 components
    • divide by 2
  • (no index) — 1 component

If this interests you, the topic is Index Notation.