Aleberto69
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- TL;DR Summary
- I'm an enginner and not a physics, however is quite a long time (years) that I try to study by myself math and physics to better undestand them deeply.. I'm quite disappointedd today as being very finichy, I alwasy end up at a point where things are not clear.. FOR EXAMPLE IN RECENT DAYS: PSEUDOVECTORS
Is it really possible to explain the concepts of vectors and pseudovectors (differencies) only using the simple mathematical structure used by authors like Griffiths (Introduction to electrodynamics), as the claim to do, without using more deep mathematics like group theory, etc?
Reading the above mentioned reference I ended up in even more confusion and disappoiting.
Please do not try to explain the topics using theory that I'm not ready as I just want to know if the approch of basics books like the quoted one is rigorous or not enough consistent to the proof.
more details:
Following their approach I 'm indeed tempted to say that the "k" base vector, being the result of "i x j" vectors is a pseudovector, and this sounds to me a contraddiction.
Furthermore the same author on applying coordinate trasformations, trasforms only vectors components and conclude that "a vector change sign" on a coordinate inversion ( ie x1'= -x1, x2'=-x2, x3'=x3). This to mee is very misleading as on a change of cordinate system you need to specify also how you like to change the vector base and, based on this choice, it will derive how components of a vector will consequently change (or not).. in any case for any choice you have had made on the new vector base, the new components of a vector will change in order that the geometrical object vector ( linear combination of base elements waighted by the proper components) will be the same.
if I apply this approach, two vectors are always the same geometrical objects regardless the coordinates system (usefull only to identify points) and vector base trasformation (and consequently the trasformation of components of the vectors) chosen and therefore their cross product do not change sign under any trasformation. And so will do their cross product. There is no place in this last way of argumenting for pseudovectors.
Many authors say ( Griffiths is one of them) that magnetic filed and angular momentum are pseudovectors and not vectors ( as they are the result of a crss product of vectors) while other authors say the contrary. Eg: Riley-Hobson-Hence in "Mathematical methods for Physics and Engineering".. pag 948 say that "It is worth nothing that, although pseudotensor can be usefull mathematical object , the description of real phisical world must usually be in term of tensors( i.e. scalars , vectors, etc) .......velocity, magnetic field strength or angular momentum can only be described by a vector and not by a pseudovector".
This last authors, later on in the text ( p949), then, however, mention about passive and active trasformation where indeed vectors behave differently.. but they are on their turn now not very clear on what that definition and assesments mean.
What I would like eventually to know is if concerning the matter of understanding many physical questions like the case of pseudovectors, the simple physics and math approach of the above mentioned references (and similar ones) is a bit naif and bluffing or if it can have a rigorous based on precise definitions and operations. In case of negative answer I would like to ask what level of mathemathical theory/methods is really consistent on consistently treating such concepts.
At the time being I do not need demonstrations based on superior mathematics ( group theory, etc) as I'm not ready for them.
I would appreciete just possible rigorous demonstrations based on simple vectors algebra definition or the comment that such a structure is not enough to describe this kind of complexity
Thank you in advance for your replies
Reading the above mentioned reference I ended up in even more confusion and disappoiting.
Please do not try to explain the topics using theory that I'm not ready as I just want to know if the approch of basics books like the quoted one is rigorous or not enough consistent to the proof.
more details:
Following their approach I 'm indeed tempted to say that the "k" base vector, being the result of "i x j" vectors is a pseudovector, and this sounds to me a contraddiction.
Furthermore the same author on applying coordinate trasformations, trasforms only vectors components and conclude that "a vector change sign" on a coordinate inversion ( ie x1'= -x1, x2'=-x2, x3'=x3). This to mee is very misleading as on a change of cordinate system you need to specify also how you like to change the vector base and, based on this choice, it will derive how components of a vector will consequently change (or not).. in any case for any choice you have had made on the new vector base, the new components of a vector will change in order that the geometrical object vector ( linear combination of base elements waighted by the proper components) will be the same.
if I apply this approach, two vectors are always the same geometrical objects regardless the coordinates system (usefull only to identify points) and vector base trasformation (and consequently the trasformation of components of the vectors) chosen and therefore their cross product do not change sign under any trasformation. And so will do their cross product. There is no place in this last way of argumenting for pseudovectors.
Many authors say ( Griffiths is one of them) that magnetic filed and angular momentum are pseudovectors and not vectors ( as they are the result of a crss product of vectors) while other authors say the contrary. Eg: Riley-Hobson-Hence in "Mathematical methods for Physics and Engineering".. pag 948 say that "It is worth nothing that, although pseudotensor can be usefull mathematical object , the description of real phisical world must usually be in term of tensors( i.e. scalars , vectors, etc) .......velocity, magnetic field strength or angular momentum can only be described by a vector and not by a pseudovector".
This last authors, later on in the text ( p949), then, however, mention about passive and active trasformation where indeed vectors behave differently.. but they are on their turn now not very clear on what that definition and assesments mean.
What I would like eventually to know is if concerning the matter of understanding many physical questions like the case of pseudovectors, the simple physics and math approach of the above mentioned references (and similar ones) is a bit naif and bluffing or if it can have a rigorous based on precise definitions and operations. In case of negative answer I would like to ask what level of mathemathical theory/methods is really consistent on consistently treating such concepts.
At the time being I do not need demonstrations based on superior mathematics ( group theory, etc) as I'm not ready for them.
I would appreciete just possible rigorous demonstrations based on simple vectors algebra definition or the comment that such a structure is not enough to describe this kind of complexity
Thank you in advance for your replies