What is the history and significance of tensor fields in physics?

[Jorrie]

Relativity is not a thing your nominal engineer ever needs. Some engineers have a curiosity that drives them to find out what they can about the topic. Some read all the popular books and still have little 'handles' on it. Most just ignore it, unless their work somehow requires it.

There are a few engineering environments where relativity plays an important role. I can think of GPS systems designs, particle accelerators and perhaps some advanced optical systems design, especially for astronomy.

Which others are there?

- Jorrie
___________________
"Curiosity has its own reason for existence" -- Albert Einstein

 

[rdt2]

The thing is that the mathematical tools that are now common in relativistic mechanics that might have application to engineering mechanics. The increasing use of differential forms in electrical engineering and continuum mehcanics is one example. The ubiquitous use of tensors is another.

 

[Jorrie]

Tensors or not?

The thing is that the mathematical tools that are now common in relativistic mechanics that might have application to engineering mechanics. ...

True - and once one knows those mathematical tools, relativity is a breeze...

However, I found that one can understand (if not quite master) a good deal of relativity without tensors.

 

[Danger]

I found that one can understand (if not quite master) a good deal of relativity without tensors.

That's a reflection of my own experience. I have a grade 9 math level, but I can feel relativity. If you want to accompish anything with it, however, you need the educational background. It's sort of like my approach to engineering. I can design and build just about anything that I might ever need in my life, and have a few patent-pending things on the go... but if you value your life, don't ever cross a bridge that I make.

 

[AlephZero]

The thing is that the mathematical tools that are now common in relativistic mechanics that might have application to engineering mechanics. The increasing use of differential forms in electrical engineering and continuum mehcanics is one example. The ubiquitous use of tensors is another.

The basic maths behind modern computational methods in contimuum and fluid mechanics (variational principles, integral equations, etc) all predate relativity.

For example

Euler: 1707-1783
Lagrange: 1736-1813
Fourier: 1768-1830
Gauss: 1777-1855
Navier: 1785-1836
Green: 1793-1840
Stokes: 1819-1903

The practical applications of the maths were a consequence of the invention of electronic computers, not of Einstein.

 

[rdt2]

The basic maths behind modern computational methods in contimuum and fluid mechanics (variational principles, integral equations, etc) all predate relativity.

For example

Euler: 1707-1783
Lagrange: 1736-1813
Fourier: 1768-1830
Gauss: 1777-1855
Navier: 1785-1836
Green: 1793-1840
Stokes: 1819-1903

The practical applications of the maths were a consequence of the invention of electronic computers, not of Einstein.

I have no argument with what you say about numerical methods and the practical results they generate - my own field is finite element analysis. However, improvements in numerical methods seldom lead to paradigm shifts in understanding. I stick by my claim that the invention (discovery?) of tensors did exactly that. And in the light of differential forms, Stokes Theorem is seen as a special case of a broader principle.

 

[Clausius2]

Burning the poll

I'm interested but not professionally, since as an engineer I have received an education for being interested in all aspects of physics. That's why we are called 4x4 in industrial and research environments. On the contrary, I've seen so many students and professors of 'advanced' physics such as relativity theory not interested on 'low level' physics that I'm suspicious that those people who know a lot about that stuff don't have a solid basis on 'supposed' easier parts of the physics, and that is a shame.

 

[AlephZero]

It would be interesting to find out the history of the use of tensor fields in physics. Possibly the concepts were being used before the name tensor was invented and the modern notation was developed.

E.g. in continuum mechanics there's the Cauchy and Piola-Kirchoff stress tensors, and the Green-Lagrange strain tensor. I don't know what notation Cauchy, Green, etc actually used, but presumably the meaning of their notation was the same as the modern version.

 

[Jorrie]

Tensor Fields

It would be interesting to find out the history of the use of tensor fields in physics. Possibly the concepts were being used before the name tensor was invented and the modern notation was developed.

From Wikipedia: "Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and was made accessible to many mathematicians by the publication of Tullio Levi-Civita's 1900 classic text of the same name (in Italian; translations followed). In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915."
and
"Many mathematical structures informally called 'tensors' are actually 'tensor fields' —an abstraction of tensors to field, wherein tensorial quantities vary from point to point. Differential equations posed in terms of tensor quantities are basic to modern mathematical physics, so that methods of differential calculus are also applied to tensors."
http://en.wikipedia.org/wiki/Tensor"

Any other interesting references?