[EMAIL PROTECTED] wrote:
In that Sussman video he gives an interesting example of something that has
been taught (and continues to be!) millions of times in calculus
classes all over the world but is just plain wrong.....
Ever heard of the chain rule? df/dx = (df/du) * (du/dx) .
Technically this is not correct because f is either is function of x
or u but not *both*.
It is quite technically correct. If you write out the whole partial
derivative mess and then collapse the terms that go away because u is a
function solely of x, your formula is quite fine (in fact, all of the
calc books *I* have do indeed give a pointer to the fact that the chain
rule is a simplified partial derivative even if it is in a footnote).
We model at varying levels of abstraction and call them correct. This
is quite acceptable, TYVM. And, in this instance, the result is
actually correct. Be careful calling things "wrong" when they are not.
A whole bunch of electricity and magnetism is *STILL* taught by the
Heaviside-Hertz pedagogy for the late 1800's. This pedagogy effectively
assumes the existence of an ether (preferred frame of reference) *and it
still works fine* for most things. However, it fails miserably for a
few things (electric motors being one).
Occasionally, you get some crackpot who goes blathering about the
inconsistencies in modern EM theory (one of the electronics hobbyist
magazines from England I read has a fairly noisy one who blathers quite
often). There are no inconsistencies in EM--you use a different
pedagogy for things where you can't assume an ether. You jump directly
to the field formulation and relativistic four-vectors. However, that
pedagogy makes calculating simple things really annoying as you have to
carry relativistic terms around until you can cancel them out at the end
(it's not even an approximation, they really do completely cancel out).
The ability to abstract at different levels of correctness and
completeness is a hallmark of modern science and engineering. The
inability to accept different levels of abstraction and correctness is a
hallmark of mathematics. Both have their place--see: Dirac delta function.
The reason this doesn't bother anyone is that students eventually (hopefully)
learn to read what the *implied* rules are behind the *informal* mathematical
notation.
And what have *you* been smoking. Students learn what gets them through
their immediate problem. I'm not much different.
I'm not going to learn all of formal lambda calculus to understand
lisp/scheme. I'm not going to learn deep pure math to balance my
checkbook. etc.
If calculus books were required to present mathermatical statements in Scheme
or Python code that ran correctly, then they would be forced to make obvious
all this implied information.
Sorry, normal programming languages are *lousy* constructs to match to
mathematics and physics.
Why do you think Mathematica and Matlab are so amazingly popular?
-a
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