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© 2000-2020   Gérard P. Michon, Ph.D.

 Arms of Gregorio Ricci-Curbastro 
 1853-1925

Tensors

In geometric and physical applications,  it always turns out that
a quantity is characterized not only by its tensor order,  but also by symmetry.

"Peter" Hermann Weyl  (1925).
 

Related articles on this site:

Related Links (Outside this Site)

Introduction to Tensor Calculus   by  Taha Sochi  (2016-02-25).
Tensors and Relativity   by  Peter Dunsby  (University of Cape Town, 1996).
 
Wikipedia :   Tensors   |   Intrinsic definition   |   Cartesian tensor   |   Glossary of tensor theory

Books :

Videos :

"Tensor Calculus"  (2014)  by  Pavel Grinfeld :   0 | 1 | 2 | 3 | 3a | 4s | 4 | 4a | 5? | 5b | 6a | 6b | 6c | 6d | 7a | 7b | 7c | 7d | 8 | 8b | 8c | 8d | 8e | 9a | 9b | 10a | 10b | 10c | 11a | 11b | 12s | 12 | 12a | 12b | 13b | 14a | 14b | 14c | 14d | 14e | 14f | 15 |
 
Tensor products demystified (1:04:14)  by  Michael L. Baker  (2016-01-17).
 
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Tensor Calculus


(2015-01-27)   Definition of a Tensor

 Come back later, we're
 still working on this one...

Wikipedia :   Tensor Calculus


(2009-08-05)   Tensors in metric spaces
What tensors  really  are.

By definition, the scalars of a vector space are its  tensors of rank 0.

In any vector space, a linear function which sends a vector to a scalar may be called a  covector.  Normally, covectors and vectors are different types of things.  (Think of the  bras  and  kets  of quantum mechanics.)  However, if we are considering only  finitely many dimensions, then the space of vectors and the space of covectors have the same number of dimensions and can therefore be put in a  linear  one-to-one correspondence with each other.

Such a  bijective correspondence is called a  metric  and is fully specified by a nondegenerate  quadratic form, denoted by a  dot-product  ("nondegenerate" precisely means that the associated correspondence is bijective).

A metric is said to be Euclidean if it is "positive definite", which is to say that  V.V  is positive for any nonzero vector  V.  Euclidean metrics are nondegenerate but other metrics exist which are nondegenerate in the above sense without being "definite" (which is to say that  V.V  can be zero even when  V  is nonzero).  Such metrics are perfectly acceptable.  They include the so-called Lorentzian metric of four-dimensional spacetime, which is our primary concern here.

Once a metric is defined, we are allowed to blur completely the distinction between vectors and covectors as they are now in canonical one-to-one correspondence.  A tensor of rank zero is a scalar.

More generally, a tensor of nonzero rank  n  (also called  nth-rank tensor, or n-tensor)  is a linear function that maps a vector to a tensor of rank n-1.

Such an object is  intrinsically  defined, although it can be specified by  either  its covariant  or  its contravariant coordinates  in a given basis  (cf. 2D example).


(2016-01-25)   Gradients
A tensor of rank 1 in covariant form:

 Come back later, we're
 still working on this one...

Tensor Analysis  by  Pavel Grinfeld  (video 2):  The Two Definitions of the Gradient

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