There are two uses of the word scalar, one from linear algebra and another from tensor calculus.
In linear algebra, vector spaces have a field of scalars. This is where the coefficients in linear combinations come from. For real vector spaces, the scalars are real numbers. For complex vector spaces, the scalars are complex numbers. For vector spaces over any field K, the elements of K are called scalars.
But there is a more restrictive use of scalar in tensor calculus. There a scalar is not just a number, but a number whose value does not depend on one’s choice of coordinates. For example, the temperature at some location is a scalar, but the first coordinate of a location depends on your choice of coordinate system. Temperature is a scalar, but x-coordinate is not. Scalars are numbers, but not all numbers are scalars.
The linear algebraic use of scalar is more common among mathematicians, the coordinate-invariant use among physicists. The two uses of scalar is a special case of the two uses of tensor described in the previous post. Linear algebra thinks of tensors simply as things that take in vectors and return numbers. The physics/tensor analysis view of tensors includes behavior under changes of coordinates. You can think of a scalar as a zeroth order tensor, one that behaves as simply as possible under a change of coordinates, i.e. doesn’t change at all.
2 thoughts on “Tensors 5: Scalars”
I’m not sure I understand the second definition of a scalar. Temperature depends on a coordinate system on a 1d line — a reference point and a scale/units, i.e. a choice of 0 and 1. Perhaps I am missing something more subtle.
You bring up an interesting point. You’re talking about a choice of coordinates on the range — how you measure temperature — but the discussion around tensors is usually about the domain, the location at which you measure temperature.