# Trying to study Tensors (but )

• Curl
In summary, the conversation is about finding a way to make sense of Exercise 8.3 on page 19 of the notes on tensors. The conversation also discusses different approaches and resources for learning about tensors, with the recommendation to start with the post linked in the conversation. The post discusses the change of coordinate formulae for three bases of the same vector space and the transformation of coordinates for a specific type of quantity in different bases.
Curl
I'm trying to figure this out but its confusing. I'm going by some notes someone put up online:
samizdat.mines.edu/tensors/ShR6b.pdf

Look at Exercise 8.3 on page 19. I got no idea how to do this, and actually I'm not even sure what it's asking. Can anyone give me some pointers? If I just read without trying to do the exercises I don't get anything out of it.

Thanks

I can't make sense of it either. Were you able to make sense of Exercise 8.2? It seems to rely only on the fact that S and T are defined as each other's inverses. I don't see how it has anything to do with "the result of exercise 5.8". And S and T are still going to be each other's inverses even if we swap their meanings.

I also can't resist mentioning that I really dislike presentations on tensors based on the "something that transforms as blah-blah" idea. These guys are trying to avoid "difficult" mathematics, but they're just making it harder. I think it's easier to learn tensors by studying differential geometry, at least if you ignore the stuff about topology. This post would be a good place to start. (Ignore the first two paragraphs and start at "A manifold...").

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Okay do you know any other good place to start on tensors? I tried Simmonds and I think it is garbage.

I'm still searching online for notes, and will try Schaums but it seems like most of them aren't good.

I don't know a text that's really great. I picked up some stuff about tensors from the SR section of Schutz's GR book, and then learned the rest from books on differential geometry. I used Spivak at the time, along with Wald (that was a GR book), but now I think a much better choice would be Lee, "Introduction to smooth manifolds". However, I'm reluctant to recommend that you try it this way, because you might get really confused by all the stuff about topology and then blame me for wasting your time.

If you want to try, I think the post I linked to is actually a good place to start. Instead of reading a book from page 1, you could read my posts and use the books only to look up details that you want to know more about. (Get the big picture first, and fill in the details later).

Curl said:
Look at Exercise 8.3 on page 19. I got no idea how to do this, and actually I'm not even sure what it's asking

I scanned the notes. It's probably considered gauche to use linear algebra to talk about tensors, but by doing that, I formed an interpretation. I've changed his notation a little.
(This long post has many opportunities for errors. Check it!)

Three bases for the same vector space are ${\bf e} = \{e_1,e_2,e_3 \}, {\bf f} = \{f_1,f_2,f_3\}$ and ${\bf g} = \{g_1, g_2, g_3\}$.

Considering the elements of each basis to be a row vector, the change of coordinate formulae can be visualized as matrix multiplication.

Assume the following:
$${\bf f} = {\bf e }{\bf S}$$
$${\bf e} = {\bf f }{\bf T }$$

$${\bf g} = {\bf f }{\bf P}$$
$${\bf g} = {\bf e }{\bf R}$$

where each of ${\bf S}, {\bf T}, {\bf P},{\bf R}$ is a 3 by 3 matrix.
and ${ \bf T } = { \bf S}^{-1}$

The convention for writing the matrix elements is illustrated by:
${\bf S } = \begin{pmatrix} S_1^1 & S_2^1 & S_3^1 \\ S_1^2 & S_2^2 & S_3^2 \\ S_1^3 & S_2^3 & S_3^3 \end{pmatrix}$

So my version of eq 5.7 is:

$$\begin{pmatrix} f_1 & f_2 & f_3 \end{pmatrix} = \begin{pmatrix} e_1 & e_2 & e_3 \end{pmatrix}\begin{pmatrix} S_1^1 & S_2^1 & S_3^1 \\ S_1^2 & S_2^2 & S_3^2 \\ S_1^3 & S_2^3 & S_3^3 \end{pmatrix}$$

In order to be consistent the result of changing from basis ${\bf e}$ to basis ${\bf g}$ directly via the matrix ${\bf R}$ must produce the same result as changing from basis ${\bf e }$ to the basis ${\bf f }$ and then changing from the basis ${\bf f }$ to the basis ${\bf g }$. This amounts to the equality:

$${\bf e} {\bf R} = ( {\bf e}{\bf S} ){\bf P}$$. For this to hold it is sufficient that ${ \bf R } = {\bf S} {\bf P}$ which we will assume.

The section that develops eq. 6.2 demonstrates the interesting fact that the coordinates ${\bf x } = \{x_1,x_2,x_3\}$ of a vector in basis ${\bf e}$ transform to its coordinates ${\bf y}$ in basis ${\bf f}$ by the rule:
$${\bf y} = {\bf T} {\bf x}$$
where the coordinates are written as column vectors. (This is contrary what we might naively expect - namely some equation involving the matrix ${\bf S}$ ).

The section with eq 8.1 hypothesizes that there is a quantity ${\bf A }$ that has a representation as 3 coordinates $\{a_1, a_2, a_3\}$ in basis ${\bf e}$. Visualizing the coordinates as a row vector ${\bf a }$ , the rules for relating the coordinates for this type of quantity to its representation ${\bf b}$ in basis ${\bf f}$ are:

$${\bf b } = {\bf a} {\bf S}$$
$${\bf a } = {\bf b} {\bf T}$$

This representation is consistent with respect two ways of changing from basis ${\bf e}$ to basis $g$.

The first way is change from basis ${\bf e }$ to basis ${\bf g}$ directly by using matrix ${\bf R}$. The second way is to change from basis ${\bf e }$ to basis ${\bf f }$ and then to change from basis ${\bf f }$ to basis ${\bf g}$. This amounts to the equality:

$${\bf a} {\bf R } = ({\bf a}{\bf S}){\bf P}$$

Since we assumed above that ${\bf R} = {\bf S}{\bf P}$ , this equality holds.

My interpretation of exercise 8.3:

Let ${\bf M }$ be the matrix that transforms from basis ${\bf v}$ to basis ${\bf w}$. Suppose there is a quantity ${\bf A}$ who coordinates ( as a row vector) ${\bf b}$ in basis ${\bf w}$ are given as a function of its coordinates ${\bf a}$ in basis ${\bf v}$ by:

$${\bf b } = {\bf a} {\bf M}^{-1}$$

Show the test of consistency may fail.

The test of consistency would be that this equality holds:

$${\bf a}{\bf R}^{-1} = ( {\bf a {\bf S}^{-1}}) {\bf P}^{-1}$$

Although we have assumed ${\bf R} = {\bf S}{\bf P}$, this does not imply what would be needed , which is ${\bf R}^{-1} = {\bf S}^{-1}{\bf P}^{-1}$, so we should be able to find a numerical example where the equality fails.

## 1. What are tensors?

Tensors are mathematical objects that describe the relationship between different physical quantities. They are used to represent physical quantities, such as forces, velocities, and electric fields, and their direction and magnitude.

## 2. Why are tensors important in physics?

Tensors are important in physics because they provide a way to describe complex physical systems and their interactions. They allow us to analyze and predict the behavior of systems in multiple dimensions and can be used to solve problems in mechanics, electromagnetism, relativity, and many other areas of physics.

## 3. How are tensors used in machine learning?

Tensors are used in machine learning to represent and manipulate large amounts of data. They can be used to store and process images, text, and other types of data in a structured and efficient way. Tensors are also important for deep learning, where they are used to create and train neural networks.

## 4. What is the difference between scalars, vectors, and tensors?

Scalars are single values that do not have any direction or magnitude, such as temperature or mass. Vectors have both direction and magnitude, and can be represented by arrows. Tensors are similar to vectors, but they can have multiple components and can represent more complex relationships between physical quantities.

## 5. How can I improve my understanding of tensors?

To improve your understanding of tensors, it is important to have a strong foundation in linear algebra and calculus. You can also practice solving problems and applying tensors in real-world scenarios. Additionally, there are many online resources and textbooks available that can help you learn and apply tensor concepts.

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