Tensors

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Introduction

As a retired K9 math teacher I said to myself I want to understand the math language used in the Einstein field equations. This led me to the issue tensors. In my math studies in Sweden we never studied more than vector calculus. I had to repeat vector calculus and learn about tensors. This page collect what I have learned so far and what sources I used. Maybe you will find it useful too.

This in one of several math pages. You find the others in the menu above under learning math.

Tibees explains vectors and reference frames with space time diagrams in one of her youtube but you have to pay to read  the text in your own pace in brilliant.org.

I started this issue writing my page about Einsteins field equation of general relativity in http://www.kinberg.net/wordpress/stellan/einsteins-field-equations-of-general-relativity/
Tensors are “the facts of the universe” (Lillian Liebherr)

Index

 

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Vector definition

I jumped into  Dan Fleisch youtube (see below under tensor basics) but discovered that I needed a basic vector update. I found that EigenChris offers a good one in this video.

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tensors

I have seen this video where Dan Fleisch  gives a very good explanation about tensors. You need to know about vectors as I do but Dan explains vectors to  in case you forgot about these.  Vectors is just one type (rank 1) of tensors in the tensor family. It has to be seen several times to understand the indexing system.

“A student guide to vectors and tensors” is available as a ebook at books.google.it/books

There is a kind of tensor called metric tensor. I like the definition shared in mathworld.wolfram.com/MetricTensor.html

“Roughly speaking, the metric tensor g_(ij) is a function which tells how to compute the distance between any two points in a given space….,.” (Read more in wolfram.com )

“Ax is the component (that pertains to and is a perfect representation of) the x-hat vector ( ed. x-unit vector)”

(In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or “hat”:  pronounced “i-hat”” (Source . Wiki )

Unit vectors are quite well explained in https://en.wikipedia.org/wiki/Unit_vector

like“the unit vectors in the direction of the x, y, and z axes of a three dimensional Cartesian coordinate system are”

“A student guide to vectors and tensors” is available as a ebook at books.google.it/books

There is a kind of tensor called metric tensor. I like the definition shared in mathworld.wolfram.com/MetricTensor.html

“Roughly speaking, the metric tensor g_(ij) is a function which tells how to compute the distance between any two points in a given space….,.” (Read more in wolfram.com )

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Rank 2 tensor

This part that starts minute 12 is a critical point that is more difficult to digest.

  • We have here two vectors:
    area vector
  • force vector

Each vector has its component (number)

“Ayx might refers to a x-directed force in a surface whose area vector is in the y-direction.”

“This combination of 9 components and nine sets of two basis vector makes this a rank two tensor”

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Tensor definition and calculus

How you can understand the rank of a tensor is well explained by eigenchris in the video below. In his channel you find a serie of tensor calculus videos.

When he continues with calculus it become very difficult if you never have done vector calculus. I have seen this video at least three times. 🙂
EigenChris shares two good definitions of tensors.

  1. Array definition: tensor is a multidimensional grid of numbers
  2. Coordinate definition: A tensor is an object that is invariant under a change of coordinates and has components that change in a predictable way under a change of coordinates.
  3. Abstract definition: Tensor is a collection of vectors and covectors combined together using a tensor product.
  4. Tensors as partial derivatives that transform with the Jacobian matrix  (Requires knowledge of matrix calculus and the Jacobian determinant )

I see that it is good to read the math world.wolfram.com definition of tensor in http://mathworld.wolfram.com/Tensor.html :

“An nth-rank tensor in m-dimensional space is a mathematical object that has n indices and m^n coi

Each index of a tensor ranges over the number of dimensions of space. However, the dimension of the space is largely irrelevant in most tensor equations (with the notable exception of the contracted Kronecker delta). Tensors are generalizations of scalars (that have no indices), vectors (that have exactly one index), and matrices (that have exactly two indices) to an arbitrary number of indices.

Tensors provide a natural and concise mathematical framework for formulating and solving problems in areas of physics such as elasticity, fluid mechanics, and general relativity.

mponents and obeys certain transformation rules.”

 

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Tensors visualized

  1. V1=1.826 (with subscript for the index value) is a variable with co-variant components 
  2. V1=4  (with superscript for the index value) is a variable with a contra-variant components  

Multiplying P and V

  • the contra-components of a vector P
  • with the contra-variant components of a vector V

e get these matrix of 9 possible multiplications or tensors rank 2:

11= VP11  12= VP12 13= VP 13
T 21= VP21 22= VP22 23= VP 23
T 31= VP31 T 32= VP32 T 22= VP22

 

Multiplying P and V

  • the co-variant components of a vector p
  • with the contra-variant components of a vector, we get these tensors:
11 12 13
T 21 22 23
T 31 T 32 T 32

 

 

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Forward transformation

I followed Eigenchris video to learn about forward and backward transformation.

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Backward transformation

I followed Eigenchris video above to learn about forward and backward transformation.

Back ward transformation gives us the inverse of matrix F (F-1)

This can be checked by mutliplying F and B
If you multiply FB you get I (the identity matrix) So B is the inverse of F. F=B-1. Read more about inverse matrices in http://www.kinberg.net/wordpress/stellan/matrices/#issue2

Matrix multiplication

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The stress tensor

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Metric tensor

eEigenChris gives a very good explanation of what the metric tensor is. The derivation of it made by DrPhysics is in my EFE page

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Conclusions

 

 

A agnostic pluralist seeker

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