Talk:Relative scalar

Example

I need to redo the example with a region that does not include the origin. The transformation is singular there and I stupidly included it. I want to add a very similar example for scalar density but the current integration region is bad for that. Jason Quinn (talk) 20:33, 20 April 2011 (UTC)

There seems to be two errors in Elementary example section. First, if we are talking about quarter slice cylinder then the range for $y$ variable in the first integral should be from $0$ to ${\sqrt {2^{2}-x^{2}}}$ rather than from $-{\sqrt {2^{2}-x^{2}}}$ to ${\sqrt {2^{2}-x^{2}}}$. So the correct first integral should be $\int _{0}^{2}\!\int _{0}^{\sqrt {2^{2}-x^{2}}}\!\int _{0}^{2}\!f(x,y,z)\,dz\,dy\,dx=16+10\pi$. Second, for cylindrical coordinates ($r,t,h$) volume element is $rdrdtdh$ rather than just $drdtdh$. So the second integral should be written as $\int _{0}^{2}\!\int _{0}^{\pi /2}\!\int _{0}^{2}\!{\bar {f}}(r,t,h)\,rdh\,dt\,dr=16+10\pi$, which is the same value as in the first integral. Kkkmail (talk) 18:05, 3 October 2012 (UTC)

You are correct that there are errors. The region should be fixed as you noted. Regarding the integrals, there are three integrals of note:

1. $\int _{0}^{2}\!\int _{0}^{\sqrt {2^{2}-x^{2}}}\!\int _{0}^{2}\!f(x,y,z)\,dz\,dy\,dx=16+10\pi$ (Wolfram Alpha proof)
2. $\int _{0}^{2}\!\int _{0}^{\pi /2}\!\int _{0}^{2}\!{\bar {f}}(r,t,h)\,dh\,dt\,dr=12+10\pi$ (Wolfram Alpha proof)
3. $\int _{0}^{2}\!\int _{0}^{\pi /2}\!\int _{0}^{2}\!{\bar {f}}(r,t,h)J\,dh\,dt\,dr=16+10\pi$ (Wolfram Alpha proof)

The difference between the last two equations is the presence of the Jacobian. As you note, when the Jacobian is added to the integral, the sum is equalized. This proves that temperature is a relative scaler of weight 1, not an ordinary scaler. This is a subtle concept and one where I find myself easily mixed up because I'm so used to "sticking" Jacobian constants where they need to be. I have reworded the article to correct the errors and hopefully made the distinction more precise. Maybe it's too verbose now. I dunno. Jason Quinn (talk) 22:44, 3 October 2012 (UTC)

Some development

I noticed some inconsistency that I would like to discuss before editing: I don't agree with the statement in the end of introduction: "There are many physical quantities that are represented by ordinary scalars, such as temperature and pressure." More specifically, it is either temperature or pressure that is an ordinary scalar.

In a small volume dV, we have dN mol particles and according to the Ideal gas law

    p.dV = R.T.dN

that is,

    p = R.T. (dN/dV)

where the last factor is n the number density, that is,

    p = n.R.T

But n is a scalar density, that I will note n̂ :

    p = n̂.R.T

And it hurts our sense of homogeneity, as we would have a weight 1 tensor on the right hand side and weight 0 on the left hand side. hence either we give temperature OR pressure a weight 0, but the other one must have non-zero weight.

Next I advocate that it is pressure that has a non-zero weight, namely 1:

    p̂ = n̂.R.T

by noticing that

    p̂.dV = -δW

is a Work exchange element, that we can integrate. Which in turn lead us to see pressure as a work density.

Would you agree with this? Bixente5691 (talk) 14:28, 24 July 2015 (UTC)

Let's add that reminds us that The pressure for an atomic system is given by :

    p = (Σ_j (m_j v_j² + r_j · F_j))/(3V)

being a true scalar over a volume is then a relative scalar.

The question of the sign of the volume weight has also to be dealt with as a this relative scalar pressure is not a pseudo scalar pressure, given that the sign of the pressure should not change under parity.

Bixente5691 (talk) 14:12, 22 April 2019 (UTC)

References