Re: Request for review and approval: Big Bang FAQ


Ulf Torkelsson wrote:
> Let me introduce a warning here, that this posting will
> be seriously off-topic and only of interest for the very
> few on this group that are engaged in teaching physics
> at the university level.

Ok, these are the theoretical tools to investigate the origin
of the universe.

> dynamics@xxxxxxxxxxxx wrote:
> > This is becoming humorous....
> >
> > Ulf Torkelsson wrote:
>
> >> There are many people with a decent education in physics
> >>that still have a problem to really understand vectors. The
> >>basic problem is that you do not really learn how to use
> >>vectors from elementary physics books such as Halliday &
> >>Resnick, and you certainly do not learn it from the
> >>mathematicians when you take your courses in linear algebra
> >>and calculus.
> >
> >
> > Ulf, your propping up your mistaken view of vectors by claiming
> > and I quote Ulf,
> >
> > Ulf wrote, "do not learn it from the mathematicians",
>
> You are misquoting me here. Please read my complete
> sentence instead:

I read it, and was disappointed by your disrespect of mathematicians,
by making such statements. I agree that there is a difference between
the *static* tensor analysis Reimann et al developed and the *dynamic*
tensor analysis AE used in GR, but your prose is quite vague.

> >>The
> >>basic problem is that you do not really learn how to use
> >>vectors from elementary physics books such as Halliday &
> >>Resnick, and you certainly do not learn it from the
> >>mathematicians when you take your courses in linear algebra
> >>and calculus.
>
> Now, what do I mean by this? The key expression here is
> "how to use". What the mathematics professor will teach you
> extremely, and that also goes for the brilliant mathematics
> teachers that I had at university, is how to add Cartesian
> vectors, what a scalar product and a vector product is, and
> how to calculate the products for Cartesian vectors. After
> this they will go on and generalise the vector concept and
> the scalar products to cover a wide range of different spaces
> including Hilbert spaces. This gives you a beautiful theory
> that you can apply on Fourier series and the solutions of
> differential equations, which is a necessary prerequisite
> for really doing quantum mechanics.
>
> You will also be introduced to cylindrical and spherical
> coordinates in the calculus course and you will know
> everything there is to know about the use of the Jacobian
> when you change coordinates in an integral. The problem
> is that this is only half the story. What you learn then
> is only that x = r sin theta cos phi and so on, but in
> my experience the notion that you can change from Cartesian
> basis vectors to curvilinear basis vectors, whose direction
> changes from point to point, is not well-developed in
> the mathematics courses.

It's very well developed, it called *tensor analysis*!
See Spiegel's "VECTOR ANALYSIS" and an introduction
to "TENSOR ANALYSIS", I think he's good.

> In the best of worlds this concept is developed when
> the students move to the physics department and take the
> courses on mechanics and vector calculus or electromagnetic
> fields. However that requires firstly an ambitious mechanics
> course in which neither the lecturer nor the book shy away
> from discussing concepts such as the derivative of a basis
> vector, and that the students are presented with a range of
> problems that are the most easily solved in different
> coordinate systems. I think this is usually done quite
> well, but still we will always find some students that manage
> to surf through the course without really thinking about how
> a vector really works.
>
> Later on there will be the vector calculus course, and in
> this course you will be introduced to the different vector
> operators in Cartesian, cylindrical and polar coordinates,
> and many students will just take them as given and will not
> think any more deeply about them. The cure here is to expose
> the students to problems, usually integrals of vector fields,
> that are so complicated that in order to solve them the
> students have to get acquainted with how to handle the vector
> fields and re-write them in another coordinate system, and
> I am sad to say after having been teaching such courses for
> years, that this is where many textbooks let the students
> down either by only exposing them to simple problems, or
> by being so opaque that it becomes virtually impossible to
> learn anything from them.

Agree, but, see Weinberg's "Grav&Cosmo" Eq.(8.7.2) that
describes the Shapiro result, that's sufficient for reality.
I would think that is the most challenging effective integral
in "classical GR". Beyond that the integration of any sort of
metric - that is in accord with G_uv = T_uv - is a specific
study related more to unified field theory.

Regards
Ken S. Tucker

.

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