Originally Posted by **Pmb** The terms "frame dependant" and "scalar" contradict each other. By definition of the term *scalar* a quantity such as energy which is independent of any frame of reference or coordinate system. Something like proper mass is a scalar though.
Let me elaborate that whether something is a scalar or not depends on the class of coordinate systems under consideration. For example; kinetic energy is an invariant under rotation of coordinate axes which makes energy a Cartesian scalar. Energy does change under a Lorentz transformation which means that its not a Lorentz scalar. |

I think you will find that just as Physicists use a much more restriced definition of vectors than Mathematicians, so too the Maths crowd has a wider definition of scalars than just frame invariants.

In many textbooks you will find somewhere towards the beginning some kind of 'discalimer' which runs along the lines of

For the purposes of this book we will take, scalars/vectors/tensors .../widgets to mean ......

Tensors of rank 0 are scalars and I think it is better to use this term for this purpose.

But invariants are not the most common purpose of scalars.

I like this approach (due to NASA)

Multiplying a vector by a scalar has the effect of changing only the magnitude, but leaving the direction unchanged. The ouptut is another vector.

Multiplying a vector by a vector (cross product) also outputs a vector but with new magnitude and the direction changed to a perpendicular.

To change the direction (and magnitude) to any desired value you need a tensor multiplying a vector.

But there is more. A scalar is a number?

This is also officially a scalar

$\displaystyle \left[ {\begin{array}{*{20}{c}}5 & 0 & 0 \\0 & 5 & 0 \\0 & 0 & 5 \\\end{array}} \right]$

and what about this equation

speed = frequency x wavelength

Are the quantities scalars or what?

In the words of that immortal sage