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

1. May 23, 2010

### 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

2. May 23, 2010

### tiny-tim

Hi Rearden!
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.

3. May 23, 2010

### 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.

4. May 23, 2010

### clem

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

5. May 25, 2010

### 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.

Last edited: May 25, 2010
6. May 25, 2010

### 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!

7. May 25, 2010

### tiny-tim

Hi sophiecentaur!

See http://en.wikipedia.org/wiki/Tensor_density" [Broken]

Last edited by a moderator: May 4, 2017
8. May 25, 2010

### 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. . . . .

9. May 26, 2010

### Rearden

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

10. May 26, 2010

### Phrak

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.