Update on 01 March 2018, 21:27, IST: I had posted a version of this post also at iMechanica, which led to a bit of a very interesting interaction there [^] too. Check it out, if you want… Also see my today’s post concerning the idea of stress, here [^].
In this post, I am going to note a bit from my personal learning history. I am going to note what had happened when a clueless young engineering student that was me, was trying hard to understand the idea of tensors, during my UG years, and then for quite some time even after my UG days. May be for a decade or even more….
There certainly were, and are likely to be even today, many students like [the past] me. So, in the further description, I will use the term “we.” Obviously, the “we” here is the collegial “we,” perhaps even the pedagogical “we,” but certainly neither the pedestrian nor the royal “we.”
What we would like to understand is the idea of tensors; the question of what these beasts are really, really like.
As with developing an understanding of any new concept, we first go over some usage examples involving that idea, some instances of that concept.
Here, there is not much of a problem; our mind easily picks up the stress as a “simple” and familiar example of a tensor. So, we try to understand the idea of tensors via the example of the stress tensor. [Turns out that it becomes far more difficult this way… But read on, anyway!]
Not a bad decision, we think.
After all, even if the tensor algebra (and tensor calculus) was an achievement wrought only in the closing decade(s) of the 19th century, Cauchy was already been up and running with the essential idea of the stress tensor right by 1822—i.e., more than half a century earlier. We come to know of this fact, say via James Rice’s article on the history of solid mechanics. Given this bit of history, we become confident that we are on the right track. After all, if the stress tensor could not only be conceived of, but even a divergence theorem for it could be spelt out, and the theorem even used in applications of engineering importance, all some half a century before any other tensors were even conceived of, then developing a good understanding of the stress tensor ought to provide a sound pathway to understanding tensors in general.
So, we begin with the stress tensor, and try [very hard] to understand it.
We recall what we have already been taught: stress is defined as force per unit area. In symbolic terms, read for the very first time in our XI standard physics texts, the equation reads:
… Eq. (1)
Admittedly, we had been made aware, that Eq. (1) holds only for the 1D case.
But given this way of putting things as the starting point, the only direction which we could at all possibly be pursuing, would be nothing but the following:
The 3D representation ought to be just a simple generalization of Eq. (1), i.e., it must look something like this:
… Eq. (2)
where the two overlines over represents the idea that it is to be taken as a tensor quantity.
But obviously, there is some trouble with the Eq. (2). This way of putting things can only be wrong, we suspect.
The reason behind our suspicion, well-founded in our knowledge, is this: The operation of a division by a vector is not well-defined, at least, it is not at all noted in the UG vector-algebra texts. [And, our UG maths teachers would happily fail us in examinations if we tried an expression of that sort in our answer-books.]
For that matter, from what we already know, even the idea of “multiplication” of two vectors is not uniquely defined: We have at least two “product”s: the dot product [or the inner product], and the cross product [a case of the outer or the tensor product]. The absence of divisions and unique multiplications is what distinguishes vectors from complex numbers (including phasors, which are often noted as “vectors” in the EE texts).
Now, even if you attempt to “generalize” the idea of divisions, just the way you have “generalized” the idea of multiplications, it still doesn’t help a lot.
[To speak of a tensor object as representing the result of a division is nothing but to make an indirect reference to the very operation [viz. that of taking a tensor product], and the very mathematical structure [viz. the tensor structure] which itself is the object we are trying to understand. … “Circles in the sand, round and round… .” In any case, the student is just as clueless about divisions by vectors, as he is about tensor products.]
But, still being under the spell of what had been taught to us during our XI-XII physics courses, and later on, also in the UG engineering courses— their line and method of developing these concepts—we then make the following valiant attempt. We courageously rearrange the same equation, obtain the following, and try to base our “thinking” in reference to the rearrangement it represents:
… Eq (3)
It takes a bit of time and energy, but then, very soon, we come to suspect that this too could be a wrong way of understanding the stress tensor. How can a mere rearrangement lead from an invalid equation to a valid equation? That’s for the starters.
But a more important consideration is this one: Any quantity must be definable via an equation that follows the following format:
the quantiy being defined, and nothing else but that quantity, as appearing on the left hand-side
some expression involving some other quantities, as appearing on the right hand-side.
Let’s call this format Eq. (4).
Clearly, Eq. (3) does not follow the format of Eq. (4).
So, despite the rearrangement from Eq. (2) to Eq. (3), the question remains:
How can we define the stress tensor (or for that matter, any tensors of similar kind, say the second-order tensors of strain, conductivity, etc.) such that its defining expression follows the format given in Eq. (4)?
Can you answer the above question?
If yes, I would love to hear from you… If not, I will post the answer by way of an update/reply/another blog post, after some time. …
A Song I Like:
(Hindi) “ye bholaa bhaalaa man meraa kahin re…”
Singers: Kishore Kumar, Asha Bhosale
Music: Kishore Kumar
Lyrics: Majrooh Sultanpuri
[I should also be posting this question at iMechanica, though I don’t expect that they would be interested too much in it… Who knows, someone, say some student somewhere, may be interested in knowing more about it, just may be…
Anyway, take care, and bye for now…]