I Definition of Second-Order Tensor by Jim Adrian

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A second-order tensor is defined as a bilinear map that takes two vectors from a vector space and produces a scalar, emphasizing its mathematical structure rather than its representation as a matrix. The discussion highlights the distinction between tensors and matrices, noting that while matrices can represent tensors, they are not synonymous. Multiple definitions of tensors exist, all mathematically equivalent, but the thread emphasizes the importance of formal definitions found in textbooks rather than informal online sources. The conversation also touches on the confusion surrounding the terms "order" and "rank" in relation to tensors. Overall, a rigorous understanding of tensors requires a solid foundation in linear algebra and vector spaces.
  • #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?
 
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  • #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
 
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  • #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.
 
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  • #39
We have given tons of answers already!
 

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