@fresh_42
Let me explain myself. Of course, I did not mean the inner product operation and matrix multiplication to be the same.
Let's say we have two vectors (vector and covector to be precise) that we will regard as matrices with one row and one column. So, by definition of inner product we...
@fresh_42
I am afraid I am missing your point. We can multiply only those matrices that have equal numbers of rows and columns. For example we can multiply a matrix 2x3 by another matrix 3x4. But how should we multiply 2x3 by 4x4? It is not defined, is it? So, how can one refer to set of...
It says in any textbook (for example, in classical text «Theory of matrices» by P. Lankaster) on matrix theory that matrices form an algebra with the following obvious operations:
1) matrix addition;
2) multiplication by the undelying field elements;
3) matrix multiplication.
Is the last one...
I am not sure I understand your point. The analytic functions form a small and restrictive class of functions. It can be broadened by dropping some requirements imposed on class members. It gives us this sequence (incomplete, I guess, but it illustrates the basic idea):
##C^\omega \subset...
For historical reasons the hyperbola always was considered to be one of the «classical» curves. The function, obviously, does not belong to C0. Apparently, is does not fit L2 or any other Lp? What is the smallest class?
It was not about higher dimensions for me, actually. It is just an example. If one talks about 2D tangent space, he uses vectors, not complex numbers. It is probably possible to consider tangent plane as Argand plane, but no one is doing that (AFAIK). There must be a reason for that.
I can't argue with that. But I fail seeing how it answers my question, sorry. I realise the difference results of this extra algebraic structure. My question is how to understand whether this extra is helpful for solving a task (thus choosing between the two formalisms to work with the problem...
Well, let's consider a specific problem (the way I see it is still my original question, just reformulated for specific situation).
There is a 2D regular surface in Euclidean 3D space. If one talks about tangent space in any of its point, he necessarily use concept of 2D linear space. Vectors...
I know this topic was raised many times at numerous forums and I read some of these discussions. However, I did not manage to find an answer for the following principal question.
I gather one deals with the same set in both cases equipped it with two different structures (it is obvious if one...
Thank you, I checked the monograph you cited. For now it looks like the best shot. But actually, I was thinking about something deeper and more thorough (probably a journal review paper).
Some critique of basics and upshots of rational thermodynamics approach can be found in «A history of...
Looking for thorough serious comparative review of modern generalisations of classical non-equilibrium thermodynamics of continua.
I have heard about several such generalisations:
1) rational mechanics by Truesdell, Coleman, Noll.
2) extended thermodynamics by Müller (and Ruggeri).
3) the...
Do we really need concept of cross product at all? I always believed cross product to be sort of simplification of exterior product concept tailored for the 3D case. However, recently I encountered the following sentence «...but, unlike the cross product, the exterior product is associative»...
The building of theoretical mechanics can be constructed using only the first and the second derivatives (those of coordinates in case of kinematics: velocity and acceleration and those of energy in case of dynamics: force and gradient thereof). It is obviously unavoidable if one wants to deal...
It surely is. Thank you!
I would appreciate you providing me with specific reference to a monograph or a review of applications of Kubo's idea (if you have such a reference at the ready, of course)? My guess is you refer to the so-called «Thermal field theory»...
Looking through the book of abstracts for «XXI International Conference on Chemical Thermodynamics in Russia (RCCT-2017)» I came across the abstract of talk given by Peter Atkins (University of Oxford) titled «Thoughts about thermodynamics» (you'll find the whole abstract at the end of the...
@fresh_42 Will it be correct (albeit not quite rigourous, but for now I am trying to grasp the very idea) to say that transformations are automorphisms of space (set, manifold) arising in context of consideration of general linear group and subgroups thereof?
Could you please help me to understand what is the difference between notions of «transformation» and «automorphism» (maybe it is more correct to talk about «inner automorphism»), if any? It looks like those two terms are used interchangeably.
By «transformation» I mean mapping from some set...
In some sense people expressing this opinion (although I agree with opinion of @StatGuy2000 that the thread title does not accurately describe Musk's view on this, but I heard other people saying just that) are right. But it is not the whole picture.
I would suggest the following analogy: some...
@DrClaude
You are right. I should have used the verb «hope» in stead of «expect» (I was under pleasant impression of my yesterday's discovery that they opened access to «Annales de l'I.H.P. Physique théorique»; hoped the Italians followed the French).
Thank you very much for checking it for...
For the first time in my lime I am not able to find home page for well-known and respected western periodical. I am looking for the paper titled «Définition covariante des équilibres thermodynamiques» by J. M. Souriau published in Supp. Nuovo Cimento, 1, I , 4 (1966) pp. 203-216 (given the fact...