Re: Integrating both sides .. only in physics?
- From: Narcoleptic Insomniac <i_have_narcoleptic_insomnia@xxxxxxxxx>
- Date: Sat, 15 Apr 2006 01:55:29 EDT
On Apr 14, 2006 11:20 PM CT, Kenneth Bull wrote:
Throughout Calc 1 and 2, I never encountered
"integration of both sides" as a way of "doing" FTC
related things. Everything was explained wihtout this
concept.
ex:
a = v '
therefore by FTC
integral of a (from t1 to t2) = v(t2) - v(t1)
I have little experience with physics, but IIRC
v' = dv / dt
...which implies...
v' dt = dv.
However, velocity if a function of time, namely v = dx/dt
where x is the position function. Formally we should be
writing v(t) = x'(t) = d[x(t)]/dt where x(t) is the
position function.
Letting a(t) = v'(t) and integrating w.r.t. t from t_1 to
t_2 yeilds
int_{t_1}^{t_2} a(t) dt = int_{t_1}^{t_2} dv(t) =
int_{t_1}^{t_2} a(t) dt = v(t_2) - v(t_1)
...by the FTC as you stated earlier.
No need for manipulating differentials like fractions,
and no need for "integrating both sides."
Now flipping to my mechanics textbook, I am shocked
to see that the calculus is all based on "moving
differentials around" and "integrating both sides."
I believe this is because intro calculus texts, for the
most part, are concerned with the theory of calculus.
However, in mechanics texts these theoretical quantities
become models for physical quantities.
For example, 'dx' is no longer an abstract concept but
represents an infintesimal change in a real physical quantity.
I'm not saying calculus texts do not introduce these
applications of the theory; the majority of them do.
I think this discussion just goes to show that it is good
to try and learn from as many different texts as possible.
Moreover, differential calculus is an abstract construct
and mechanics is the attempt to apply this theory to
the physical reality.
Is this just to simplify the application of FTC (no
need to really understand FTC if one just plays with
differentials and integrates both sides)?
I don't have much experience with physics, but in my
opinion, yes.
Is this mechanics/physics based calculus/integration
okay?
Of course.
If so, why do the math people stay away from this stuff?
The other side of the coin would be the question, "why
do the physics people stay away from the other stuff?"
I feel like the math people and physics people should
come to some kind of agreement on all this.
Ha!
I will go out on a limb and say there exist physicists
who belive mathematicians are too formal and rigorous in
their treatise, just as there exist mathematicians who
believe physicists are too informal and sloppy with their
notations.
However, please don't take this quote out of context. I
know for a fact that there are a great deal of physicists
who are extreamly talented mathematicians and vice versa.
Regards,
Kyle Czarnecki
.
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