AlvaroDíaz wrote:
But for make this conventions, at least, there are some considerations, for example, can ask for some coherence. If sin(2 kg) have sense, in the functional definition for sin, you have this:
The problem with your argument is that [y] is not (generally) equal to [x]/kg. Let's revert to the standard [t] = s, [x]=m, [y]=m, dx/dt = y is equivalent to {dx}[x]/{dt}[t] = {y}[y] == {dx}/{dt} m/s = {y} m. Clearly there is a problem here, as whilst no one is going to argue with the numerical value equivalence, metres-per-second is not equal to metres dimensionally.
This highlights one of the major problems with creating a system of quantities that includes extra dimensions, and is effectively the same problem we have with many legacy formulae - there is a hidden (very assumed) conversion factor needed to dimensionally balance the equations.
In this case, keeping it simple by letting [y] = [x] = m, the dimensional form of the equation should be (assuming that speed is the desired quantity😞
dx/dt = k•y
= {dx}/{dt} m/s = {k}[k]•{y}m, where {k} = 1
reducing dimensionally to
= m/s = [k] m
which gives
[k] = 1/s
Replacing [t] with kilograms rather than seconds trivially gives the [k] = 1/kg
Stuart
Philip Oakley wrote:
StuartBruff wrote:
So a function like f(x):= 2•x2 + 3•x3 would be forbidden because it would try to add m2 to m3?
Essentially yes.
OK. Then I believe it's wrong. One of the very first equations I was taught in physics was the famous relationship between distance and time for a moving object:
So here we have a function of t that involves multiple powers of t and yet returns a dimensionally consistent result.
.....
However the taylor expansion part can be used to show that, when viewed as trigonometry, the sin/cos/tan of angles reguires an angle dimension depending on how you view it.
The Taylor expansion is often used to "show" this. Unfortunately, the process usually just takes the dimensionless (mathematical / numeric) form of the Taylor expansion and forgets how it is derived. A back-to-basics derivation from the differentials shows that the Taylor expansion is effectively unit neutral.
Stuart
Yep.
From Stuart's file: https://www.ptcusercommunity.com/servlet/JiveServlet/download/86229-11630/Test%20-%20Units%20Angles%2002%20extract%202.m… There are a margin note about SiUnitsOf, instead UnitsOf. Here not a magical unit appear, but preserve the argument unit. I guess that is not correct, but a good approach. I guess that the most correct option is introduce new dimensions: this is ln(m) as new one, ln(kg), sin(m), etc.
AlvaroDíaz wrote:
StuartBruff wrote:
The problem with your argument is that [y] is not (generally) equal to [x]/kg. Let's revert to the standard [t] = s, [x]=m, [y]=m,
Hi Stuart.
This is not the case, the standard here isn't [t] = sec, it's just a geometrical parameter, not the time.
OK ... my own assumption here - show a physicist with the notation dsomething / dt in the context of a discussion involving "standards" and there is rather a tendency to think of "t" as time (ISO 80000-3:2007 3-7 symbol t = time). 
.... What the picture shows is that t can have another dimension than the one that make [t]^2=1, at least with this definition of sin and cosin. Further t is an angle. For mechanics, where t is the time, you never take sin(t), but sin(w*t), where w is the angular speed. If you see some equation in mechanics with sin(t), and t as time, it's implicit a factor of 1 rad/sec multiplying t. That's what's actually happen with the mechanical interpretation of parametric curves: they are traveled at one rad/sec. But if you need them at some other speed, need to introduce another value as the omega factor.
Putting aside the discussion about whether trig functions (or any function) can take dimensioned arguments, I was addressing the fact that the differential equations as you presented them are not dimensionally consistent. Using quantities of dimension 1 simply hides this dimensional inconsistency rather than addressing it. The equations *require* the dimensioned scaling constant to make them consistent. When added, the "Can''t Be" expression in your derivation becomes a "Can Be"
Stuart
Hi Stuart.
That's true, but what you prove it's that sin and cos can't get dimensions, like any other unary function in physics or math. For example, the minus unary function, or the factorial or gamma functions. Why do you expect that after apply -(5 sec) they go to be transformed to - 5 meters? Same thing with x or y, which goes to be applied to some argument, can't make some units to be appear after their application.
Best regards.
Alvaro.
Philip Oakley wrote:
Alvaro,
There is a BIG problem with these trig functions in that the broader community does not agree what they define or how they are defined.
Some argue that f(x) = x - x^3/3! +x^5/5! - ... defines the sine function, while others argue that sine (angle) = Opposite / Hyponeuse of a plain plane 'tri-angle' (that's three angles 😉
Sure, for this reason I state that one must to say about which definition of the function one is taking about. The fact that sin can be developed in such way is using reals, but units actually don't need those numbers, units need only rationals. See my old answer to you document here: Re: Why is Mathcad so Keen on Units? . There, also, there are a Mona's post saying that take in consideration this document for further develop in mathcad (2010 year). If we have that, the ability to introduce new dimensions in mathcad, we probably are playing introducing a dimension like ln(distance) with some unit like ln(meter). I don't se in my old ruler calc ln(x/cm), I see only ln(cm):
Best regards.
Alvaro.
AlvaroDíaz написал(а):
I don't see in my old ruler calc ln(x/cm), I see only ln(cm):
Best regards.
Alvaro.
Welcome to Re: Nostalgia - a slide rule
Philip Oakley wrote:
StuartBruff wrote:
.. but you cannot calculate the sine of 2kg.
I disagree ... must get round to writing up it up one day ...
Hi Stuart,
Ah, the misunderstandings and bad groundings of Quantity Calculus.
Unfortunately, Quantity Calculus, as it is promolgated makes the error of splitting the arithmatic from the 'dimensions' (note I said arithmatic rathers than mathematics;-). This wrenches the whole units, dimensions and scalings away from being a well defined and integrated system.
http://web.ist.utl.pt/ist12219/data/60.pdf Two alternative derivations of Bridgman’s theorem, Mario N. Berberan-Santos and Lionello Pogliani, Journal of Mathematical Chemistry 26 (1999) 255–261
Those people prove something more strong than any unidimensional unit must to be write as a*[x]^b, they prove that any univariate function must to be write as f(x) = a*x^b ! Check de demonstration, please. That's make engineering much more easy.
Best regards.
Alvaro.
AlvaroDíaz wrote:
Those people prove something more strong than any unidimensional unit must to be write as a*[x]^b, they prove that any univariate function must to be write as f(x) = a*x^b ! Check de demonstration, please. That's make engineering much more easy.
I think you missed one hidden point. They showed that when 'x' has dimensions, then by the "principal of dimensional homogeneity", the function of 'x' must be of that form.
The reverse is not true. if x has no dimensions (i.e. pure number) then it can be a polynomial. (see the footnote about Descartes)
The other part that isn't explored is the multivariate case where we can have the case (usually for Length, see Descartes) where we can get the form f(x,y) = k.(x^2 + y^2), but it is (must be) homogeneous for the situation to be valid. (which, via Descartes) leads to the Angle vs Length, vs number of D/dimensions fun!).
Hi Philip.
When the dimensions of the coefficient is burried in all that dimension "1" stuff, then the dimension information gets lost.
Correct math set up the correct dimension for each coefficient:
AlvaroDíaz wrote:
Hi Philip.
When the dimensions of the coefficient is burried in all that dimension "1" stuff, then the dimension information gets lost.
Correct math set up the correct dimension for each coefficient:
"Correct math set up the correct dimension for each coefficient:"
*That* is exactly what I've been saying all along (ie, over a decade). If we have to have to have special rules for dealing with a particular quantity (eg, angles), then (IMO) there is something fundamentally wrong with the corresponding quantity calculus (where the word calculus is used in a generic not familiar ("diff and int") sense).
The important fact that is lost in most "Proofs by Taylor Expansion" is what the Taylor series is and where it comes from - that is, a means for determining the coefficients of a function's power series representation.
The correct starting point is to assume that some function can be represented as a power series and, to be dimensionally (ie, physically) consistent, the coefficients *must* balance the powers.
I fail to see what inherent difference it makes to the physics of the situation whether we have,
f(x) = 3 x
s(t) = s0 + u.t + 1/2.g.t^2
s(t) = a0 + a1.t + a2.t^2 + a3.t^3 + ...however many other derivatives are considered necessary to model the behaviour, which might very well turn out to be equivalent to ...
s(t) = sin(t) (in it's series expansion form that I couldn't be bothered to type in ... )
Everything else gives the appearance of "magic" ("Here Be Dragons (angles), ... and they don't care about physical consistency, so we just pretend they're dimensionless do some pure maths, and then we'll add the dragons (angles) back in again and everything will be OK!").
Stuart
