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In mathematics, scalars are defined as elements of a field. Are scalars in mathematics invariant under coordinate transformation? In physics, the way they are defined, I know they are. Since in physics, they are defined to be quantities having only magnitude and no direction, they are invariant under rotations or translation. Some sources say that in mathematics, scalars are, by definition, invariant under coordinate transformations. Can anyone please explain how defining scalars as elements of a field would intrinsically assume invariance under coordinate transformation? If not, can you give an example of a field whose elements may not be invariant under coordinate transformation?

wordman soft
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  • When physicists call something a “scalar quantity” I think they mean like “a physical quantity that would make sense to represent by a number because its value doesn’t depend on which coordinate system we’re using.” And a “vector quantity” in physics is like “a physical quantity with magnitude and direction that would make sense to represent by a math vector because its magnitude and direction don’t depend on the coordinate system we’re using.” – littleO Jul 25 '23 at 09:34
  • Yes, but what about mathematicians? There are two separate articles in wikipedia- Scalar(physics) and Scalar (mathematics). The first line of Scalar(physics)-In physics, scalars are physical quantities that are unaffected by coordinate system transformation. But in Scalar (mathematics), I found nothing such mentioned explicitly. It mentions instead-A scalar is an element of a field which is used to define a vector space. So does this definition inherently imply that scalars are invariant under coordinate transformation? Or is such requirement not necessary at all for scalars in mathematics? – wordman soft Jul 25 '23 at 09:45
  • I’m not even sure what it would mean in math for an element of the scalar field of a vector space to be “invariant under coordinate transformation”. A vector space has an associated field of scalars, that’s all. – littleO Jul 25 '23 at 09:54
  • If $f:\mathbb R^n \to \mathbb R$ is a smooth function and $x , v \in \mathbb R^n$, you could prove a theorem to the effect that the value of the directional derivative $D_v f(x)$ is invariant under coordinate transformations. That’s an example in math where we have a numeric value and we might want to prove that the numeric value is in some sense invariant under coordinate transformations. – littleO Jul 25 '23 at 09:59
  • Please do not look for the meaning of words by digging through wikipedia articles. Work on a few concrete examples to see why vector fields transform differently from "scalar" fields, or however one wants to call those. – Kurt G. Jul 25 '23 at 10:18
  • @littleO So is it that, though it wouldn't mean much in math for an element of the field of a vector space to be “invariant under coordinate transformation”, they are still in some sense invariant under coordinate transformation? – wordman soft Jul 25 '23 at 11:44
  • Hmm, I’m still not sure how to make that into a precise statement and I think I’d just not even try. A vector space has an associated field of scalars, that’s all. If we introduce a new coordinate system, of course the value of any particular element of the field of scalars doesn’t change. It’s only physicists who, when representing some physical quantity by a number, have to stop and think, “Wait a second, are we sure the value of this number doesn’t depend on which coordinate system we’re using?” – littleO Jul 25 '23 at 11:56
  • A mathematician might prove that the value of the dot product of two vectors doesn’t depend on which orthonormal basis we’re using, for example. Mathematicians could prove other theorems like that — what about the divergence of a vector field, for example. A “geometric” quantity such as the divergence of a vector field should hopefully be invariant in some sense under coordinates changes — it shouldn’t matter which coordinate system we’re using. – littleO Jul 25 '23 at 12:00

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