[Phys-L] Re: Elecric Field Lines

[Carl Mungan <mungan@USNA.EDU>]

I'm coming into the discussion late, but FWIW:

> It is one of those liberties which we take with mathematical rigor, as eg;
> replacing (ds/dt)*dt  with ds and calling it "canceling" the dt.

I don't see the analogy. In this case, you *did* cancel the dt. I
specifically tell my students that "dt" isn't zero unless you're in a
math class. In physics class, dt is a small change in t. (Or a
differential could be a small amount of something, eg. charge dq.)
How small? Small enough that physical quantities are constant on that
scale - eg. v = dx/dt because the speed of the car is constant during
that time interval, rho = dq/dV because the charge density is
constant in that volume. If not, chop finer. But at some point
(before I reach atomic or quantum scales!) I've chopped fine enough.
That's the difference between the math world where discontinuities
can occur and the physical world. So in physics dt is finite and I
cancel it. I think other ways of thinking are unnecessarily pedantic
for physics undergraduates. I'd like them to be bold in their use of
calculus without having to revert to epsilon-delta proofs every time.

|| Just yesterday I had every student in a junior E and M class tell me
|| that, if Cvector dot Avector = Cvector dot Bvector for any Cvector
|| then we can cancel the Cvector on both sides of the equation and
|| determine that Avector = Bvector.

And that answer is correct if Cvector is nonzero and we're working in
one dimension. Now if we're working in three dimensions, we'll extend
the argument and show we need three linearly independent Cvectors.
For simplicity, we'll use the three Cartesian unit vectors, one per
dimension.

(I know we all know all this. I'm just emphasizing that we *can*
start with the students' answer and build from there. And we only
need three Cvectors not an infinite number.) -Carl
--
Carl E. Mungan, Asst Prof of Physics  410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-5002
mailto:mungan@usna.edu     http://usna.edu/Users/physics/mungan/
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