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Thread: Proof: Physical Dimensions behaving as algebraic quantities

  1. #1 Proof: Physical Dimensions behaving as algebraic quantities 
    Forum Junior AndresKiani's Avatar
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    Is there a rigorous proof that explains why physical dimensions behave as algebraic quantities. This concept allows Physicists to incorporate dimensional analysis in their calculations and theories. I'm interested in a proof or rigorous argumentation to why this works. Any help would be great.


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    I acknowledge that there may not be a formal proof for this, but perhaps a satisfying explanation or any form of rigor would be great.


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    I wonder if one defines a sub dimensional space as one where one of the higher dimensions is frozen as a constant?

    So do you have to work downwards? First assume your n-dimensional space and then "carve out" your series of lesser numbered dimensions?

    I assume you are talking about dimensional spaces that actually describe physical and spatio-temporal scenarios?

    Or is it a purely mathematical process you are wanting?

    Edit: did you see my earlier thread from last month ?

    Dimensions
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    Quote Originally Posted by geordief View Post
    I wonder if one defines a sub dimensional space as one where one of the higher dimensions is frozen as a constant?

    So do you have to work downwards? First assume your n-dimensional space and then "carve out" your series of lesser numbered dimensions?

    I assume you are talking about dimensional spaces that actually describe physical and spatio-temporal scenarios?

    Or is it a purely mathematical process you are wanting?

    Edit: did you see my earlier thread from last month ?

    Dimensions
    I'm looking for a purely mathematical argumentation. I think I can prove just why algebraic operations work on these species by thinking about functions that map length to area for example. However, I think that one could imagine of a better argumentation involving more abstract mathematical structures like spaces and fields.
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    Quote Originally Posted by AndresKiani View Post

    I'm looking for a purely mathematical argumentation. I think I can prove just why algebraic operations work on these species by thinking about functions that map length to area for example. However, I think that one could imagine of a better argumentation involving more abstract mathematical structures like spaces and fields.
    Is there anything of interest in this other thread I started on another forum?
    Surfaces in higher dimensions - Mathematics - Science Forums
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    Quote Originally Posted by geordief View Post
    Quote Originally Posted by AndresKiani View Post

    I'm looking for a purely mathematical argumentation. I think I can prove just why algebraic operations work on these species by thinking about functions that map length to area for example. However, I think that one could imagine of a better argumentation involving more abstract mathematical structures like spaces and fields.
    Is there anything of interest in this other thread I started on another forum?
    Surfaces in higher dimensions - Mathematics - Science Forums
    No, that thread isn't helpful in this discussion. The definition of a surface in mathematics isn't relevant here, at least I don't see a relevancy... I'm also on that forum, my username is Symmetrica.
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    Quote Originally Posted by AndresKiani View Post
    Is there a rigorous proof that explains why physical dimensions behave as algebraic quantities. This concept allows Physicists to incorporate dimensional analysis in their calculations and theories. I'm interested in a proof or rigorous argumentation to why this works. Any help would be great.
    i am not sure if i fully understand your question, but i'll give it a try:
    in the SI system of measurement, you have basic dimensions, from those you can form derived dimensions, using those in a measurement gives a standard format quantifier x unit of measurement.eg:
    pressure=force acting on a surface: 1Pa=1N/m^2
    kinetic energy is force acting over a distance : E=F x s, J=N x m (a)
    force is acceleration acting on a mass: F=m x a, N=kg x m/s^2 (b)
    plug-in (b) in (a)
    J=kg x (m/s)^2, E=m x v^2
    replacing v (velocity) by c (celeritas,latin for velocity used for speed of light and speed of sound)
    E=mc^2
    however, applying mathematical methods, you get:
    E=1/2 mv^2
    "Of course today every rascal thinks he knows the answer, but he is deluding himself" (A.Einstein)
    guilty as charged
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    Quote Originally Posted by AndresKiani View Post
    Is there a rigorous proof that explains why physical dimensions behave as algebraic quantities. This concept allows Physicists to incorporate dimensional analysis in their calculations and theories. I'm interested in a proof or rigorous argumentation to why this works. Any help would be great.
    I assume that your question is about things like 1 kg + 2 kg = 3 kg. If so, ask yourself what the meaning of 1 kg + 2 uvulas might be.

    The equals sign requires commensurate quantities on both sides. That includes especially the units.
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    Quote Originally Posted by tk421 View Post
    Quote Originally Posted by AndresKiani View Post
    Is there a rigorous proof that explains why physical dimensions behave as algebraic quantities. This concept allows Physicists to incorporate dimensional analysis in their calculations and theories. I'm interested in a proof or rigorous argumentation to why this works. Any help would be great.
    I assume that your question is about things like 1 kg + 2 kg = 3 kg. If so, ask yourself what the meaning of 1 kg + 2 uvulas might be.

    The equals sign requires commensurate quantities on both sides. That includes especially the units.
    I think it's more about why a Newton is a kilogram × metre / second˛ rather than something like log(kilogram) × emetre / sin˛(second)
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Quote Originally Posted by AndresKiani View Post
    Is there a rigorous proof that explains why physical dimensions behave as algebraic quantities. This concept allows Physicists to incorporate dimensional analysis in their calculations and theories. I'm interested in a proof or rigorous argumentation to why this works. Any help would be great.
    I thought maybe this relates to the Original Poster's question...

    The following is a geometric proof demonstrating BLDC motor parameters behaving as algebraic quantities which can be represented geometrically.

    Geometric Depiction:

    Suppose I have a 100kv (100 rpm per applied volt) BLDC motor with a fixed volume of copper winding and I measure the winding resistance lead to lead as 1 ohm, but I am free to change the length and cross section of the copper winding, while retaining the same copper volume. The geometric proof demonstrates that in this scenario, changes in KV (rpm per volt) are proportional to the square root of changes in conductance. For example suppose the original winding is 1 unit length and 1 unit cross section area. Now I divide the winding length by sqrt(3) [1.73205], new length = 0.5773 and I multiply the cross section by sqrt(3) [1.73205], new cross section area = 1.73205 to keep the same copper volume. The resistance drops to 1/3 of the original value while the conductance increases by a factor of 3. When this happens, as depicted in the geometric proof, the KV increases by a factor of sqrt(3) [1.73205], new kv value = 173.205 (square root of the conductance change factor with fixed copper volume). Another case of an increase in conductance x 3 leading to a change factor of KV of 1.73205 is keeping the same number of turns (same conductor length and volume) but changing the "termination" of the windings from Wye to Delta.

    This has been previously posted elsewhere at: http://vedder.se/forums/viewtopic.ph...=705&start=130



    The "traditional" formula for describing changes in KV is as follows (no change in termination):

    KN=C
    K=C/N

    K = kv = new kv (max rpm per volt) no load
    N = turns = new # of wire turns per tooth
    C = constant = original kv x original # turns

    Changing termination wye to delta increases kv by a factor of 1x1.73205 [sqrt(3)]
    Changing termination delta to wye decreases kv by a factor of 1/1.73205 [1/sqrt(3)]

    ^but notice "turns," "constant," and "termination" are not SI base unit derived variables.

    I have written a new formula that gives identical results using only SI derived variables (no turns, constant or termination variables).

    My formula for describing changes in KV is as follows:

    D=sqrt(E/(V*N))
    E=N*V*D^2
    V=E/(N*D^2)
    N=E/(V*D^2)

    D = Change Factor of KV (rpm/v)
    E = Change Factor of Conductor Resistivity (ohm-meters)
    V = Change Factor of Conductor Volume (meters^3)
    N = Change Factor of Conductor Resistance (ohm)

    For example, changing termination wye to delta, change factor of Resistance is 1/3 (conductance change factor 1x3) while the change factor of KV is 1.73205.

    Another example is shortening the winding length by a factor of 1/sqrt(3) and increasing the cross section by a factor of 1x1.73205... also leading to a change factor of Resistance is 1/3 (conductance change factor 1x3) while the change factor of KV is equal to 1.73205.
    Last edited by devin-m; December 28th, 2017 at 09:54 PM.
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    Quote Originally Posted by devin-m View Post
    Quote Originally Posted by AndresKiani View Post
    Is there a rigorous proof that explains why physical dimensions behave as algebraic quantities. This concept allows Physicists to incorporate dimensional analysis in their calculations and theories. I'm interested in a proof or rigorous argumentation to why this works. Any help would be great.
    I thought maybe this relates to the Original Poster's question...

    The following is a geometric proof demonstrating BLDC motor parameters behaving as algebraic quantities which can be represented geometrically.

    Geometric Depiction:

    Suppose I have a 100kv (100 rpm per applied volt) BLDC motor with a fixed volume of copper winding and I measure the winding resistance lead to lead as 1 ohm, but I am free to change the length and cross section of the copper winding, while retaining the same copper volume. The geometric proof demonstrates that in this scenario, changes in KV (rpm per volt) are proportional to the square root of changes in conductance. For example suppose the original winding is 1 unit length and 1 unit cross section area. Now I divide the winding length by sqrt(3) [1.73205], new length = 0.5773 and I multiply the cross section by sqrt(3) [1.73205], new cross section area = 1.73205 to keep the same copper volume. The resistance drops to 1/3 of the original value while the conductance increases by a factor of 3. When this happens, as depicted in the geometric proof, the KV increases by a factor of sqrt(3) [1.73205], new kv value = 173.205 (square root of the conductance change factor with fixed copper volume). Another case of an increase in conductance x 3 leading to a change factor of KV of 1.73205 is keeping the same number of turns (same conductor length and volume) but changing the "termination" of the windings from Wye to Delta.

    This has been previously posted elsewhere at: code to change the motor amp limit - Page 14 - vedder.se forums



    The "traditional" formula for describing changes in KV is as follows (no change in termination):

    KN=C
    K=C/N

    K = kv = new kv (max rpm per volt) no load
    N = turns = new # of wire turns per tooth
    C = constant = original kv x original # turns

    Changing termination wye to delta increases kv by a factor of 1x1.73205 [sqrt(3)]
    Changing termination delta to wye decreases kv by a factor of 1/1.73205 [1/sqrt(3)]

    ^but notice "turns," "constant," and "termination" are not SI base unit derived variables.

    I have written a new formula that gives identical results using only SI derived variables (no turns, constant or termination variables).

    My formula for describing changes in KV is as follows:

    D=sqrt(E/(V*N))
    E=N*V*D^2
    V=E/(N*D^2)
    N=E/(V*D^2)

    D = Change Factor of KV (rpm/v)
    E = Change Factor of Conductor Resistivity (ohm-meters)
    V = Change Factor of Conductor Volume (meters^3)
    N = Change Factor of Conductor Resistance (ohm)

    For example, changing termination wye to delta, change factor of Resistance is 1/3 (conductance change factor 1x3) while the change factor of KV is 1.73205.

    Another example is shortening the winding length by a factor of 1/sqrt(3) and increasing the cross section by a factor of 1x1.73205... also leading to a change factor of Resistance is 1/3 (conductance change factor 1x3) while the change factor of KV is equal to 1.73205.
    Wholly irrelevant to the opening post, and also you already posted it elsewhere, please be more circumspect with your posting.
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