Damn you mathematicians

[Daisydog]

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Well, if you want to pull a fast one on the students, you'll just write it as theta = s/r and they'll see that the units cancel. I think that textbooks tend to do that anyhow. I vaguely recall it being first introduced like that in HRK.

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This is the definition of a radian - the angle such that the arc length subtended by that angle on a circle is equal to the radius.

Boro - I see what you're saying. But I think if you just explicitly mention that radians are essentially just 2*pi times a fraction of a circle, it becomes pretty clear that they are dimensionless, AND that this is what you'd want for expressing lengths.

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Yes, I know. Perhaps I'm not making myself clear. The problem that I have with it is this:

meters = meter radians

The radians magically vanish when I have told the students to explicitly and carefully handle all units. And they only vanish under certain circumstances; only when converting from angular to linear measure, never when converting from one system of angular measure (radians) to another (degrees or revolutions, for example).

As a consequence, one can apparently magically put radians whereever one likes. My new house is going to be 3000 square footradians. My car gets 25 radianmiles/gallon on the highway, and 21 miles/gallonradian in the city.

Now if you'll excuse me, I'm going to go scramble up a couple of eggradians for my lunchradian.

[img]/images/graemlins/tongue.gif[/img]

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Let X = number of poker tournaments you play
Let Y = number of poker tournaments you win
Let P = the percent of poker tournaments you win

Then Y = X*P

tournaments = tournament percents

Oh my god! The units aren't working out!

Now if you'll excuse me, I'm going to go scramble up a couple of eggpercents for my lunchpercent.

[holmansf]

I agree that the metric carries units, but g_ij is not the metric. The metric is g_ij dx^i dx^j. You need the differentials to make it a 2-tensor- otherwise you just have an array of functions.

When we're dealing in the Euclidean space (and thinking of it as modeling a plane in which you measure distances) the coordinates (x,y) actually are measuring the distance in certain directions from a reference point, and thus have units.

The formula

ds^2 = r^2 dtheta

is saying that the round metric on the circle is equal in two different coordinate systems. On the left you have the coordinate given by arclength from some reference point. In this system the coordinate has units meters since it is actually measuring a distance. On the right you are using the angle as a coordinate. Consider a different coordinate system on (a part of) the circle given by arclength squared: t = s^2 (obviously this is only a valid coordinate system away from the reference point from which arclength is measured). Then:

dt^2 = 4s^2 ds^2 => 1/(4s^2) dt^2 = ds^2.

Now you're saying 1/(4s^2) has units m^2. By a similar argument you could get almost any function of s to have units m^2.

Are you just saying that when writing metrics you should should ignore any units on the coordinates and then write, for example, m^2 after the metric? This doesn't seem like a good idea since the whole point of dimensional analysis is to check your formulas (which relate different coordinates) using units. If you just define that all g_ij have the same coordinates it will naturally be consistent, but you're just throwing out all information about the units of the coordinates.

[Metric]

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I agree that the metric carries units, but g_ij is not the metric. The metric is g_ij dx^i dx^j. You need the differentials to make it a 2-tensor- otherwise you just have an array of functions.

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Obviously all reference too coordinates goes away if you express the metric in coordinate-independent fashion -- in that case, its just a geometrical object "g". But here we're talking about going back and forth between coordinate systems, in which case g_i_j are the components of the metric tensor, carrying appropriate units to keep everything under control and comprehensible.

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When we're dealing in the Euclidean space (and thinking of it as modeling a plane in which you measure distances) the coordinates (x,y) actually are measuring the distance in certain directions from a reference point, and thus have units.

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I disagree -- you're taking an example of a particular coordinate system where the metric has a very simple form, and then claiming that the coordinate differentials have certain dimensions based on the fact that you can almost get away with pretending the components of the metric aren't there.

Once again, in every coordinate system, the length element takes the form ds^2 = g_i_j dx^i dx^j -- it doesn't matter if its cartesian, polar, or whatever else you dream up. The left hand side is a length only -- the right hand side is made up of coordinates and the metric.

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The formula

ds^2 = r^2 dtheta

is saying that the round metric on the circle is equal in two different coordinate systems. On the left you have the coordinate given by arclength from some reference point.

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NO! The left hand side is a length only!

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In this system the coordinate has units meters since it is actually measuring a distance.

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I have no idea why you're dreaming up such a contorted system. The metric handles all dimensional analysis in any coordinates -- you don't need two coordinate systems (one for coordinates, and other to describe "arc length") to write down a simple distance, which is just a scalar quantity.

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On the right you are using the angle as a coordinate. Consider a different coordinate system on (a part of) the circle given by arclength squared: t = s^2 (obviously this is only a valid coordinate system away from the reference point from which arclength is measured). Then:

dt^2 = 4s^2 ds^2 => 1/(4s^2) dt^2 = ds^2.

Now you're saying 1/(4s^2) has units m^2. By a similar argument you could get almost any function of s to have units m^2.

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You're confusing yourself by naming one coordinate "s", which is also the lable used for the arc length. I will call it instead "x" and call its units "nats" (because it is a very "natural" coordinate).

So your first coordinate system is simple:
ds^2 (meters)^2 = 1 (meters/nats)^2 dx^2 (nats)^2

Now you define a new coordinate "t" whose units I will call "nuts". t (nuts) = s^2 (nats^2) Note that NO DIMENSIONS OF DISTANCE ARE YET INVOLVED. Nats and nuts are dimensionless units, like radians and degrees.

We start out with g = 1 (meter/nat)^2 in the first coordinate system, and then transform it to the new one g' via the standard rule:

g' = dx/dt dx/dt g

Now adding units, we note that the new metric is:
g' (meters/nuts)^2 = (dx/dt)^2 (nats/nuts)^2 g (meters/nats)^2

I.E. g' = 1/4t (meters/nuts)^2

And thus the length element in the new coordinate system is:
ds^2 (meters)^2 = 1/4t (meters/nats)^2 dt^2 (nats)^2

Please tell me that you see it now.

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Are you just saying that when writing metrics you should should ignore any units on the coordinates and then write, for example, m^2 after the metric? This doesn't seem like a good idea since the whole point of dimensional analysis is to check your formulas (which relate different coordinates) using units. If you just define that all g_ij have the same coordinates it will naturally be consistent, but you're just throwing out all information about the units of the coordinates.

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Where in your example did we lose any information??? You can use "dimensionless units" all you want -- radians, degrees, nats, nuts, etc. etc. etc. and keep yourself consistent with those. They carry information, but they do not carry dimensions of length -- nor do any coordinate differentials. By no means should you ever have a coordinate differential magically turn into something with dimensions of distance without involving the metric -- that is what it is for!

[Metric]

Made one typo:

And thus the length element in the new coordinate system is:
ds^2 (meters)^2 = 1/4t (meters/nats)^2 dt^2 (nats)^2

should read:

And thus the length element in the new coordinate system is:
ds^2 (meters)^2 = 1/4t (meters/nuts)^2 dt^2 (nuts)^2

[jstnrgrs]

I haven't read the whole thread, but I agree with boro here. The solution here is to abandon the radian unit entierly and just to treat angle measurements as pure numbers.

As a consequence, the degree unit would be equal to pi/180.

Trig functions would take pure numbers as their arguments. This wouldn't be a problem since, acording to what I said about degrees above, sin(30degrees)=sin(pi/6).

Just eliminate the term radian, define the term degree as being equal to pi/180, and there are no more problems.

[holmansf]

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I have no idea why you're dreaming up such a contorted system. The metric handles all dimensional analysis in any coordinates -- you don't need two coordinate systems (one for coordinates, and other to describe "arc length") to write down a simple distance, which is just a scalar quantity.


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I didn't dream up anything. This is the problem we were talking about. Remember Borodog's formula:

s = r*theta

s and theta are different coordinates on the circle of radius r, and this is the formula that relates them. I understand now that in physics you often write ds^2 for the metric. Serendipitously, in these coordinates the round metric on the circle actually is ds^2.

I think we will have to agree to disagree on this issue. However, let me just say that I don't understand the formula

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ds^2 (meters)^2 = 1 (meters/nats)^2 dx^2 (nats)^2


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Where does the last nats^2 come from?

Also, I appear to have made a typo in the last paragraph. It should have been, "If you just define that all g_ij have the same units...." (not same coordinates) Maybe that makes it clearer.

[Metric]

dx^2 has dimensionless units of nats^2 just as dtheta^2 has dimensionless units of radians^2 or degrees^2, depending on your choice of coordinate system.

But this is not my thread and I can only show how a system is consistent in so many ways before I get bored with dimensional analysis. If you want to believe that coordinates have dimensions of length "except for when they don't," be my guest. I doubt the future of democracy hangs in the balance.

[Mano]

Can't you just explain it using units if you choose?

s (units of length) = r (units of length / radius) * theta (radians) * (1 radius/radian)

[holmansf]

The object of dimensional analysis is to check your answers. Refering to your nat/nut example, your final answer for the metric in t coordinates is:

ds^2 = 1/4t dt^2

By your reasoning 1/4t should have units meters^2/nuts^2. However it appears to only have units 1/nut (or meters^2/nut?) since t has units nuts. This is the information you have lost by defining g_ij to have appropriate units in all cases. Note that when you replace nats by meters (and nuts by meters^2 since x^2 = t) the dimensional analysis does work to confirm that the answer is correct.

You're example just boils down to the fact that if in the formula

g dx^2 = g dx/dt dx/dt dt^2

you "cancel the dt's" it works out. This is obvious (and thus your system is consistent), but it in no way helps you check your answers. I too tire of explaining dimensional analysis, and will therefore not post in this thread anymore.

[surftheiop]

Am i getting leveled or dont the units work out ?
tournaments won = tournies played * (tournieswon/100 tournies played)
tournies played cancels and it seems units work out for me?