The journey

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Tensorland is a place where we believe your Tensor Calculus education can begin in first grade when you start learning Addition.  Homework problems in grade school and high school can be engineered to make you think about things that are important in Tensor Calculus.

Addition builds on Counting.  In Tensorland we take the view that we are not adding numbers, we are adding quantities.  For our purposes here anything can be a quantity if you can count it.  I don't want to add 3+4=7.  Instead I want to say I have 3 quarters in one pocket and 4 quarters in another pocket and that means I have 7 quarters.

We take a particular interest in drawing a link between length and value.  For every value there is a corresponding length and for every length there is a corresponding value.

As a hint of what is to come, imagine that something can hold two numbers and we can add examples of it together:

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

Multiplication builds on Addition.   If we are working with length values then multiplication gives us areas.

It is important to do multiplication on the units as well as the numbers.  

• (2)(50) =100 by itself is just not enough.
• (2 hours) (50 miles/hour) = 100 miles is satisfying.

Don't give me 100.  Give me 100 miles because the amusement park is 110 miles away and that means we are close!

Algebra is the study of doing math with symbols that hold numbers.  Quick note, these symbols are often called variables and we might say that a variable holds a value instead of saying that a variable holds a number.  For now value and number are close enough to be "same thing".

Algebra is a language we developed to better express what we think when we notice trends in numerical relationships.  Algebra makes extensive use of equations and equality.

We can agree that everything discussed in that high school algebra textbook has importance, but we wish to point out a few of those ideas are light-years higher in importance.

Some of the math concepts you learn in Algebra will reappear when you study Vector Spaces in Linear Algebra.  Those math rules will take you to challenging and beautiful ideas in Tensor Calculus.

As a final hint, Mother Nature loves "linear".  Oh, and also...

• y=mx is linear.
• y=mx+b is not.

Geometry is the study of shapes.

The kind of vector we like is sometimes called a geometric vector.  A geometric vector has a magnitude and a direction.  We make it a point to say "geometric vector" because sometimes the word vector is used for something that is very general (very few rules).  

Trigonometry, a subset of Geometry, is the study of the side lengths and angles of a right triangle.  In Tensorland we make the choice to put all of Trigonometry under the umbrella of Geometry.

It's already been said that the most important rules of algebra show up in the definition of a Vector Space.

Vector Space must include identity elements for addition and multiplication.  We can use "multiply by one" to do work that preserves Invariance.

First you learn what it takes for a set and two operations to qualify as a Vector Space.  After that you will learn there is another space called the Dual Space.  It gets crazy at this point and we will bring in both the Mad Hatter and the Hookah-smoking Caterpillar to make the discussion more interesting.  We will be looking at vectors and one-forms and after learning how one-forms are different from vectors you will learn that you can put your vector in the Dual Space--and in fact, you want to do that.  But still, it's crazy, quite crazy, that those one-forms, which are different from vectors for reasons that are important--are vectors. 

We will follow the advice of Alice and use lots of pictures.

You are required to follow strict rules when you build a tensor.

You see this when we tell you that (4,3) is not a vector unless each number is attached to a directed length segment.  The number has to come from a difference--that 4 could have come from 5-1 or 12-8.  Granted, it's ok for it to come from 4-0.  We're happy if you comply with all the above because it guarantees that your vector is in some way a movement from an initial point to final point.  All of this screens out points that have no place in our work.