Definition of Second-Order Tensor by Jim Adrian

  • Context: Undergrad 
  • Thread starter Thread starter jamesadrian
  • Start date Start date
  • Tags Tags
    Tensor
Click For Summary
SUMMARY

A second-order tensor is defined as a multi-linear map from two vectors in a vector space V and its dual V* to a scalar, which can also be represented as a two-dimensional matrix. The rank of the tensor is the sum of the number of vectors it maps, with a second-order tensor having a rank of 2. The discussion emphasizes that while matrices can represent tensors, they are not equivalent to the tensor itself. Various definitions exist, but they are mathematically equivalent, and formal definitions are best found in textbooks rather than online sources.

PREREQUISITES
  • Understanding of vector spaces and dual spaces
  • Familiarity with multi-linear maps
  • Knowledge of tensor ranks and orders
  • Basic concepts of general relativity and Einstein's equations
NEXT STEPS
  • Study the mathematical definition of tensors in "Introduction to Tensor Analysis and the Calculus of Moving Surfaces" by G. Strang
  • Explore the concept of multi-linear maps in linear algebra
  • Review the role of tensors in general relativity, particularly the Einstein tensor
  • Examine the differences between tensors and matrices in "Linear Algebra Done Right" by Sheldon Axler
USEFUL FOR

Mathematicians, physicists, and engineering professionals seeking a rigorous understanding of tensors, particularly in the context of general relativity and advanced mathematics.

  • #31
jamesadrian said:
Transformation rules must be explicitly stated in a formal definition of a second rand tensor or any tensor.

Who came up with this meaning of "formal definition"? Are you incapable of deducing the transformation rules from the rank of the tensor and which indexes are upper and which are lower?
 
Physics news on Phys.org
  • #32
PeterDonis said:
You mean you are unable to fill in the dots from what has already been said? Posts #5 and #7 in particular?

That is correct.

Jim Adrian
 
  • #33
jamesadrian said:
That is correct.

Then this thread's level cannot be "A", since you do not have that level of background knowledge. Changing it to "I".
 
  • #34
jamesadrian said:
That is correct.

Do you understand what vector spaces and their duals are? Do you understand how vectors and dual vectors (or covectors or 1-forms--nomenclature varies among different sources) transform under transformations of the coordinates?
 
  • #35
PeterDonis said:
Then this thread's level cannot be "A", since you do not have that level of background knowledge. Changing it to "I".

You didn't ask Why.

Jim Adrian
 
  • #36
jamesadrian said:
You didn't ask Why.

Why what?

If you mean I didn't ask why you can't fill in the dots, I don't care why. The mere fact that you can't means you don't have "A" level background knowledge in the topic of this thread.
 
  • #37
If anybody here has an answer to my question, they can write to jim@futurebeacon.com

Jim Adrian
 
  • Sad
Likes   Reactions: weirdoguy
  • #38
jamesadrian said:
If anybody here has an answer to my question, they can write to jim@futurebeacon.com

If you would answer the questions I asked in you post #34, we would be able to gauge your level of background knowledge. I don't know how you expect anyone to be able to give you an answer you can understand without that information. If you had had an "A" level of background knowledge the answers already given in this thread would have been more than enough.

If you are unwilling or unable to provide that information, we'll just close this thread.
 
  • Like
Likes   Reactions: pervect and vanhees71
  • #39
We have given tons of answers already!
 

Similar threads

Undergrad How to derive irreducible representations/tensors?
  • · Replies 22 ·
Replies
22
Views
4K
Undergrad The Order and Valence of Tensors
  • · Replies 6 ·
Replies
6
Views
4K
Graduate MTW Gravitation: Exercise 5.1 | Beginner GR Stress-Energy Tensor Symmetry
  • · Replies 15 ·
Replies
15
Views
2K
Undergrad Invariance of a tensor of order 2
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Deriving tensor transformation laws
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Transformation of second rank tensor
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Explore Coordinate Dependent Statements in Orodruin's Insight
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad Understanding the Stress-Energy Tensor in Special Relativity
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Pressure term in the energy stress tensor
  • · Replies 9 ·
Replies
9
Views
3K
Graduate Find SEM Tensor from Lagrangian of Temporal Variable
  • · Replies 7 ·
Replies
7
Views
1K