Re: Request for review and approval: Big Bang FAQ

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.

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:

>>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.

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.

Ulf Torkelsson

.

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