pmb_phy said:
Hi Rob
I'm curious about this magnitude thing. While I do not disagree with you on this part, I do have a Dover text called Differential Geometry, by Erwin Kreyszig, in which the author defines a vector according to the simple definition I gave above. I gave the correct definition in parentheses for those more informed on the matter. In your opinion, what would be your guess as to why the author would define a vector in this way with no mention of a metric??
I this instance I chose that definition so as not to bring in higher order of mathematics that the OP may not understand.
Thanks
Pete
Classic tensor calculus usually emphasized geometrical quantities by their invariance under transformations, probably in the spirit of Felix Klein's Erlangen Program:
"every system of geometry deals only with such relations of space as remain unchanged by the transformations of its group." Upon specifying the group of transformations (e.g., Euclidean, Lorentz, conformal, projective, etc..), you may be implicitly defining a specific metric structure.
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I don't have a copy of Kreyszig's Differential Geometry handy, although I did use it as my supplementary text for the first differential geometry course I took. I do have Kreyszig's Advanced Engineering Mathematics (5th ed, 1983), which possibly has a similar philosophy on this issue. A.E.M., Chapter 6 is called "Linear Algebra I: Vectors".
From (A.E.M., 6.1 "Scalars and Vectors"),
"A directed line segment is called a vector. Its length is called the length of the vector and its direction is called the direction of the vector. Two vectors are equal if and only if they have the same length and direction.
The length of a vector is also called the Euclidean norm or magnitude of the vector."
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Here, the emphasis on the Euclidean length of a vector is probably based on familiarity with what is generally taught at the elementary level. However, as I usually argue [see below], the most important feature of a vector is the parallelogram rule. A quantity can have a magnitude and a direction... but without the parallelogram rule, it's not a vector.
Here is how I use the directed-line-segment concept to describe vectors.
I draw a directed-line-segment (an "arrow") and refer to its size (rather than "magnitude" which is already claimed by standard definitions and which involves additional structure) and direction. At this point, there are no numbers assigned to either the size or the direction. Indeed, without additional structure, there is no way to compare two directed-line-segments in the plane... unless the direction of the two directed-segments coincide [in which case, one can express one as a scalar multiple of the other].
In spite of not having any additional structure [and independent of any structure that might be imposed], one can, however, still do the parallelogram rule. The parallelogram rule is fundamental to the notion of a vector.
If one now introduces a metric, one can now form numbers from vectors. Indeed, coordinate-free vector notation [including the abstract index notation] practically enforces this idea: given v^a, one cannot form a scalar without additional structure. A metric g_{ab} allows one to form a scalar g_{ab}v^av^b. (Graphically, a metric can be introduced by specifying (for example) a circle centered at the base-point.)
Rather than introducing the metric directly, one can introduce a group of transformations-of-coordinate-systems which can preserve the scalar nature of g_{ab}v^av^b, then possibly determine what that g_{ab} must be to be compatible with that transformation. Thus, by specifying that the group is the "ordinary rotation group", you implicitly introduce the Euclidean metric... "Lorentz transformations"... Minkowski metric, etc... (Note, however, that there are groups of transformations that will not introduce a metric.)
