This question really pertains to motivating why vectors have components whereas scalar functions do not, and why the components of a given vector transform under a coordinate transformation/ change of basis, while scalar functions transform trivially (i.e. ##\phi'(x')=\phi(x)##).(adsbygoogle = window.adsbygoogle || []).push({});

In my more naive days, coming from a physics background, my earliest introduction to vectors and scalars was in terms of so-called "Euclidean vectors", having the properties ofmagnitudeanddirectionand having the pleasing intuitive picture of arrows pointing in particular directions. With this in mind, it always made perfect sense to me why vectors have components when represented with respect to a given basis, since one needs, in general, more than one coordinate to describe a particular direction with respect to a given coordinate system. The basis vectors (induced by the coordinate system) "point" along each of the coordinate axes, and so by describing a vector in terms of its components along each of these basis vectors (i.e. how much of the vector "points" along each coordinate axis) one can capture the direction in which the vector points (with respect to the particular coordinate system chosen). Furthermore, a scalar has a magnitude, but no direction associated with it, hence it can be fully described by a single number at a given point in space and does not require components with respect to a basis. In terms of a scalar function, one simple specifies a coordinate system and for each set of coordinate values within this system, the function has a single value.

When it comes to basis transformations, one seems to usually consider coordinate bases, induced by a particular choice of coordinate system, so under coordinate transformations the components of a vector will change. Again, this makes sense, since a vector has an independent existence from any particular basis, with its direction and magnitude being intrinsic properties. Hence, under a coordinate transformation the vector itself must remain unchanged, which requires that its components change in an appropriate manner. Furthermore, since a scalar function (evaluated at a particular point) is just a number, with no directional dependence, its value at a particular point should remain invariant under any given coordinate transformation, since its value should not depend on an (essentially arbitrary) choice of coordinates.

This was all well and good (if I've understood it correctly), until one considers vector spaces in an abstract sense. Here, a vector is simply an element of a given set that satisfies a prescribed set of axioms, and a scalar is simply an element of the underlying field, ##\mathbb{F}## associated with the given vector space, ##V##. One can use any basis one likes to describe vectors within a given vector space, and a given vector can be described in terms of its components with respect to the basis vectors of a particular basis. Since all (finite) ##n##-dimensional vector spaces are isomorphic to a (##n\times##) Cartesian product of their underlying field, ##\mathbb{F}^{n}## one can represent a vector as a ##n##-tuple of scalars with respect to a given basis. Of course, the vectors themselves exist independently of any given basis, and so again one ends up with an appropriate transformation law for vector components under a change of basis. So far, so good - this tallies up with the previous notions of vectors in a general sense.

The problems arise for me, in that one can have vector spaces in which there is no notion of a vector having a direction, or a magnitude. Although, having said this, I get that, intuitively, by describing a vector in terms of its components with respect to a given basis, each component quantifies "how much" of the given vector "points" along the direction of each of the basis vectors.

Furthermore, a set of functions can be considered as a vector space, and so any given function can be described in terms of its components. How does one argue that, in general, if one chooses a particular basis (or coordinate system) a vector is then described in terms of its components, whereas a scalar (or a scalar function) simply has a value and no components?

Essentially, my questions are:

1. What is the general argument for why one describes vectors in terms of their components (with respect to a given basis), and why does a scalar not have components? (Are they simply defined that way? Is the point that a scalar is fully specified by a single value, i.e. a magnitude, and therefore does not require additional information of coordinates, since this would imply that it has components along particular directions, and thus a direction associated with it. Whereas, to fully specify a vector quantity, one must describe its magnitude and direction, and to do this with respect to a given basis requires one to specify "how much" of the vector is "pointing" along each of the basis vectors, and this is quantified by the components of the vector along each of the basis vectors?!)

2. In physics, scalars are described as having magnitude, but no direction, hence they are invariant under rotations of coordinate systems, but why should they be invariant under (essentially) arbitrary coordinate transformations?

3. Following up on questions 1 and 2, if one considers a (scalar) function, why does this not have components, and why should it remain invariant under general coordinate transformations, i.e. why should it transform as ##\phi'(x')=\phi(x)##?

(I have some ideas on the answers to these questions, but would like to here other people's opinions on the matter).

Sorry for the long-windedness of this post, I'm hoping to convey the point that I'm at in my understanding to aid anyone who might answer. Any help would be much appreciated.

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# I Vector components, scalars & coordinate independence

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