Is mass density a scalar or a scalar density of weight -1?

[Rearden]

Hi,

I'm a little confused as to the nature of mass density. I've always seen it referred to as a scalar. Now by conservation of mass, when you integrate mass density over a volume, you get a scalar quantity. But volume transforms like a scalar density of weight 1, so shouldn't mass density transform like a scalar density of weight -1?

Thanks,
Rearden

 

[tiny-tim]

Hi Rearden!

… mass density. I've always seen it referred to as a scalar.

I've always seen volume referred to as a scalar, even though (as you say) it's actually a scalar density.

I think this is because people are almost always interested in whether something is scalar as opposed to vectorial (or tensorial etc) …

in that sense, volume is a scalar, and so is (mass) density.

 

[sophiecentaur]

Neither quantities are vectors (that goes without saying), so they must be scalar quantities. The difference is that Mass is an extensive variable and Density is an intensive variable.

 

[Meir Achuz]

Hi,

I'm a little confused as to the nature of mass density. I've always seen it referred to as a scalar. Now by conservation of mass, when you integrate mass density over a volume, you get a scalar quantity. But volume transforms like a scalar density of weight 1, so shouldn't mass density transform like a scalar density of weight -1?

Thanks,
Rearden

Yes, but +1 or -1 doesn't matter because 1/(-1)=-1.

 

[Phrak]

I've been confused by the given answers (and question) for some time, clem and Tiny and
sophiecentaur. Mass density in R^3 appears to be a well-behaved scalar. Under a general linear coordinate transformation it remains constant; it doesn't pick-up a 'density' coefficient. Mass, on the the other hand, is coordinate system dependent, and is not a simple scalar, so that in going from m to cm, say, it picks up a factor of 1000. Tensor density values are additive, so mass density has a tensor density of 0. Volume has a tensor density of 1, and mass therefore has a tensor density of 0+1=1.

The keywords seem to be density, extention, Jacobian (determinant), and tensor.

 

[sophiecentaur]

Can I have a reference to "mass density" from someone, please?
Or does it just refer to things like kg m^-3?

I have a feeling that this conversation may be at a higher level than I initially thought and I don't want to appear any more dumb than necessary!

 

[tiny-tim]

Hi sophiecentaur!

See http://en.wikipedia.org/wiki/Tensor_density"

 

[sophiecentaur]

Owch.
I did tensors in prehistoric times (1964) and managed to answer the questions correctly but none of us 'got' them, in my group.
I'll get my coat. . . . .

 

[Rearden]

Sorry, I wasn't thinking hard enough about integration...I can see why it's an honest scalar now.
Thanks everyone!

 

[Phrak]

I've been confused by the given answers (and question) for some time, clem and Tiny and
sophiecentaur. Mass density in R^3 appears to be a well-behaved scalar. Under a general linear coordinate transformation it remains constant; it doesn't pick-up a 'density' coefficient. Mass, on the the other hand, is coordinate system dependent, and is not a simple scalar, so that in going from m to cm, say, it picks up a factor of 1000. Tensor density values are additive, so mass density has a tensor density of 0. Volume has a tensor density of 1, and mass therefore has a tensor density of 0+1=1.

The keywords seem to be density, extention, Jacobian (determinant), and tensor.

Haha, you'll be back tomorrow sophiecentaur. Without your post I wouldn't have learned about 'intensive' and extensive'.

But wait a minute! Have I got this upside down and backwards? If I have a space filled with a substance having a density of 5 kg/m³, and change to centimeters, the value of the scalar changes from 5 to 5000, or kg/cm³.