Tensor Calculus

---

 The following discussion is intended as a preview for Tensor Calculus.  It will briefly mention why we want to have tensors.  It will talk about non-Euclidean geometry and apologize for the fact that when you are studying tensors you also have to study non-Euclidean geometry.  Two examples will be provided to introduce you to the strangeness which deserves Star Trek music.  Invariance will be described briefly and informally using a story of a gymnast’s balance beam.  The discussion will then lightly and irreverently poke fun at how what happens with regard to coordinates.  We will declare our independence from them and after that, it seems we can’t stop talking about them–they get as much as 80% of the talk time.  Some of the “how do you know you can do that?” questions have the answer “you can’t”.  

The math that gives the answers was developed after we found the answers.  Think of the song “White Rabbit” by Jefferson and the part where the White Knight is talking backwards.  That’s what happened.  Some would argue that this wasn’t limited to Tensor Calculus but that discussion is beyond our scope. 

Tensor Calculus finds Linear Algebra extremely useful and isn’t shy about stealing every tool to be found in the Linear Algebra tool shed.  Tensor Calculus made adjustments and improvements (every change required a green light from the physical data).  We can appreciate the way we are allowed to move things around when working with Tensor notation.  We don’t see this in Matrix Multiplication, except for cases where all the matrices in a set are diagonal.  

Tensor Calculus includes a provision: there may be times that you CAN take what you got using Tensor Calculus, and translate it back into Matrix Algebra.  There is something we do with the tensor calculus notation that noticeably increases the work required when typing it, but that extra work lets us see how we should put things in order prior to doing the “tensor to matrix” conversion.  We also argue that the extra work is helpful to students learning the notation.

There is more content to learn but we believe that should be put in a sequel movie.

Why do we want tensors?

We might begin by asking, why do we need tensors?  We might then define tensor calculus as the sum of the math knowledge you need to make tensors do those things.  For instance, we might point at a particular matrix in physics and say “we need a tensor to transform that matrix”.  You can then understand why a book about tensor calculus spends a lot of time talking about transformations.  

We can do quite a bit in physics with vectors and we can use matrices to do necessary things to vectors.

There are places in physics where we need matrices, and there are times we need something “bigger” than a matrix to do something mathematical to a matrix.

You might say, “well, build something with numbers that is bigger and can do the job.”

Well, there’s more to it than just putting numbers together in some way that builds a supermatrix.  The need for tensors came about because physics was advancing, and with those advancements we had to make changes to how we play the math games.  In a way we broke down a door and when we walked through it we left behind some of the rules that Euclid believed would always be true.  You have to learn this post-Euclid world along with learning to build the supermachines of tensor calculus.  At some point you might be looking at something that has lots of numbers and the gods have decided that this isn’t “scary” enough so you have to answer several questions regarding the “structure” that goes with your numbers.

Euclid in the Rearview Mirror

Everything you were taught in Geometry that was attributed to Euclid made sense.  It’s fair to ask for an example of something that puts Euclid’s geometry in our rearview mirror.

Imagine that two of us could start at the equator, both walking north on paths that seem parallel but as both of us keep walking north, we slowly get closer together and our paths intersect at the north pole.  To explain this and other things that are strange, we are told that reality as Euclid described it was “flat” and that to go further in physics, you must deal with realities that are “curved”.  To discuss this curvature, while saying as little as possible, we might choose to say that the curvature shows up when we compare one system to another. 

If you want an example, we offer a story of two ladybugs.  One is at the north pole of a sphere and the other is at the center of the sphere.  They each agree to walk on a path that takes them to the same point at the equator of the sphere.  They move in a synchronized way so that when the “north pole” ladybug is at (x,y) the “center” ladybug is at (x,0).

• If we ask “north pole” to move at a constant speed, “center” will report a velocity that is getting slower and slower.
• If we then repeat the trip and ask “center” to move at a constant speed, the “north pole” will report a velocity that is getting faster and faster. 
• 
  

We might decide that, in a way, it is unfair: the study of tensors must teach you what these changes mean, in addition to teaching you to use the “math machines” that we call tensors.

Invariance

A gymnast works on a balance beam.  We say that the width of the balance beam is invariant.  The gymnast will probably tell you that the width of the balance beam is 10 centimeters.  It won’t matter if someone else comes along and tells you the width is 100 millimeters.  

There are an infinite number of widths that we might choose for the measurement and they correspond to an infinite number of values that might take the place of the 10 and 100 seen previously.  From these we have an infinite number of pairings (number, unit of measurement) that together give us 10 centimeters.  Knowing this you can work out math such that:

• If you are given a number, you can calculate a measurement width.
• If you are given a measurement width, you can calculate a number.

We can consider that the number and the measurement width are conjugate one to the other.  This notion of conjugacy appears repeatedly in the story of Tensor Calculus.

Invariance considers vectors to be good friends.  Vectors will do the things that physics says must be done (addition and scalar multiplication).  A vector is a difference or a set of differences.  Those differences will easily move from one system to another (example: a set of length differences in inches can be converted to a set of length differences in centimeters).

If we turn to geometry where Alice insists that we draw things, we see the addition of vectors but if we stare at two points we realize there is no answer to the question “how do you add those two points together?”

Independence from Coordinates

The laws of physics are independent of coordinate systems.  That doesn’t mean that we don’t ever want to use coordinates–we could probably say that the only time we don’t use coordinates is when we want to show you that it is possible.

(a paraphrase) Tensors offer us coordinate-independent descriptions for laws of geometry and laws of physics.

Given how much we use coordinates, we sympathize with students finding it more difficult to gain traction with the idea.  We hope the following helps.

We argue that displacement is independent of coordinates.  A displacement of two miles is what it is and it doesn’t matter if it came from an odometer reading of 99,994 to 99,006 or from an odometer reading of 100,003 to 100,005.  

Half a dozen vehicles drove from the school to the amusement park and all of them are reporting different coordinate numbers for start mileage and finish mileage.  None of those numbers matter, we just care about the 120 mile displacement from the school to the amusement park.  That number will tell us it takes 2 hours for the trip, assuming an average velocity of 60 mph.

Now contrast all of the above with a scenario where we say the answer depends on the odometer reading.  Two friends who bought the same car have a mechanical breakdown that is covered by a 100,000 mile warranty.  One car has an odometer reading of 99,994 and the other car has an odometer reading of 100,003 when they arrive at the dealership.  What happens next is so horrific that I will not tell it to you.  The background music was "Oh Fortuna" by Mozart.

The Why of Transformations

It might help to discuss why we would want to transform coordinates (example: you are in Cartesian coordinates and you decide you want Polar coordinates):

• We may want to be in Cartesian Coordinates to do some math but we want to end up in Polar Coordinates because “r” gives you the vector’s magnitude and “theta” gives you the vector’s direction.  
• We may want to adjust coordinates to make a calculation easier.  The first coordinate system may show (4,3) and a rotation changes that to (5,0).  The change might make half the terms in a summation go to zero (love, love, love).  
• We may decide it is advantageous to change to new units so that the speed of light is 1.  

One lecturer was kind enough to be candid about it: “we admit there is a coordinate system there, it’s just that we keep things general by not saying which one it is.”  

Stealing Tools from Linear Algebra 

Tensor Calculus steals every tool that was developed by the good people of Linear Algebra.

1. Basis Sets, Basis Vectors, Orthonormal Basis Vectors
2. Vectors, Vector Spaces
3. Matrices, Matrix Addition, Matrix Multiplication
4. Dot Products
5. Cross Products
6. 
   

Linear Algebra alone was not enough.  You will learn things that give it an upgrade:

• Index Notation
  • It’s a game, you need to learn all the rules and then you can start learning why the rules work.  
    • A cautionary note: you may end up going to math textbooks to see what the math people say about a topic and that’s were it gets really weird; you may read an author who tells you that we can add points together.
• Contraction
  • We had contraction in matrix multiplication when A(mn)B(np) gave us C(mp) and you notice that the ‘n’ disappeared; it disappeared because of contraction.
  • 
    

Oh, and we should be honest about another theft: several things needed for Tensor Calculus come from Set Theory and Group Theory.  Our goal is to tell you as little as possible, and yet tell you enough that you know where to go to read more about a topic.

Variations of Tensor Notation 

This part is fingernails on the chalkboard.  You might read one author who does most of the rules we find and you learn to do it that way, and then you find another author who “breaks” one or more rules that you learned.

If you read through several authors and do a comparison you will conclude that if a tensor has two indices (letters like ‘i’ or ‘j’) on it, there are four possible locations for their placement.  

• The two locations in the superscript can be called northwest and northeast.
• The two locations in the subscript can be called southwest and southeast.

It is skewed though so that the southwest position will be further to the east than the northeast position.  

The “why” for having the four positions and doing the skewing is so that we can correctly place a first tensor with one index on it next to a second tensor with two indices on it.  The “one index” will be either on the left or the right of “two indices”.  The notation below isn’t the actual way it looks but we can somewhat show the idea: the second choice is wrong because we see “iji” and we like “iij” for having the two i symbols adjacent one to the other.

1. a(i)B(ij)
2. B(ij)a(i)

Back to the idea of authors not following rules.  One author will explain that for a particular something you should have one index “up” and the other index “down” (“up” and “down” are slang for superscript and subscript).  Another author writes both indices in subscript but somehow avoids a wrathful punishment.  We are grateful to lecturers and writers who explained that some physicists say the choice was acceptable since we were in a particular story where we were in a flat Euclidean space and the math needed to flip the upstairs or downstairs status of an index is something equivalent to “multiply by 1”, thus no change happens.

What to Anticipate in the Next Movie

At the start of the second movie we will tell you to grab your butterfly net and to collect several morphisms.

• Homomorphism
• Isomorphism
  • After catching homomorphism catching this second creature will be easy
• Homeomorphism
  • This idea might be difficult to catch.  If we have a function that is differentiable then we have a homeomorphism between a first set of x values and a second set of the corresponding f(x) values.
• Diffeomorphism
  • A diffeomorphism is an isomorphism between two differentiable manifolds
  • The above explanation makes it sound easy–no, it is challenging.  We respect you for your resolve as you push through it like a snow plow.
  • 
    

You noticed that we kept using the word “differentiable”.  The white knight invites you to join him on a journey back in time to when you first learned about differentiation.  The two of you will take things that you know now and turn Differentiation from what you learned into something better, something that rocks!

It might seem that we are getting paid to differentiate every math object we can find.