Quote:
Originally Posted by NotPrinceHamlet
Has anyone got a good reference for what the mathematical operation of the tensor direct product, i.e. http://upload.wikimedia.org/math/e/9...dfc2c9a876.png does?
I've got an ok book
http://www.amazon.com/Brief-Tensor-A.../dp/038794088X
and its all going along well, with me hoping to get enough maths under my belt to tackle GR, and then suddenly the author slips in that circled cross symbol near the end of a chapter with almost no preamble or definition of what operation it represents. Almost as if he thought I wouldnt notice!
As a result, suddenly after racing through the book I've hit a brick wall at that point - all was going swimmingly until then! 
I'm particularly interested in developing a feel for *what* it does, not just its strict definition of what it is. |
It's the operator for the tensor
outer product as opposed to the inner product. The inner product is simply the dot product. (For vectors
a.b= ab cos(A).)
Consider a tensor of rank 1 which is a vector. A vector is defined as having both magnitude and direction. The number of directions defines the rank.
The outer product of two vectors is a tensor of rank 2.
For example in 2-dimensional space consider vectors
U and
V.
U = u1
a + u2
b and
V= v1
a + v2
b a and
b are basis vectors.
The outer product
UV = u1v1
aa + u1v2
ab + u2v1
ba + u2v2
bb
Each component of magnitude uivj now defined in 2 directions (
aa,
ab,
ba,
bb).
UV is therefore a tensor of rank 2.
The outer product of 2 tensors results in a tensor which is the sum of the ranks of the 2 tensors.
Hope this simplifies matters.
Steven