Re: help: first year calculus applied to intro EE
- From: Brian VanPelt <bvanpelt@xxxxxxxxxx>
- Date: Fri, 10 Aug 2007 23:58:19 -0400
On Fri, 10 Aug 2007 17:20:31 -0700, Robert Israel <israel@xxxxxxxxxxx>
wrote:
On Aug 8, 10:49 pm, Brian VanPelt <bvanp...@xxxxxxxxxx> wrote:
On Thu, 09 Aug 2007 00:04:10 -0500,Robert Israel
<isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
TTroy <tine...@xxxxxxxxx> writes:
I am pretty familiar with first year calculus (including FTOC and
convergence of improper integrals).
I am now reading an EE circuit theory book where there is a common
calculus step in several derivations that I can not follow 100%.
For example:
i(t) = dq(t)/dt (i=current is time rate of change of q=charge)
Then book would say, therefore....
q(t) = int[from -inf to t] i(t)dt
I hope it's assuming somewhere that the limit of q(t) as t -> -infinity is 0.
This happens a lot in the book, where the "reverse of a derivative
formula" results in an improper integral. Now, it makes sense when I
think of things like this as "the charge that passed the point from
beginning of time to current time (where negative infinity is
beginning of time)," but what exactly is the calculus step taken? I
mean, how do we get an improper integral from the original derivative
formula (what manipulation or theory or reasoning do we use to do that
step)? What about issues of convergence?
Basically, what allows me to do such a step?
z(t) = dy(t)/dt ==> y(t) = int[from -inf to t] z(t)dt
(assume only that y(t) is differentiable on the interval considered)
It's not true then...
The Fundamental Theorem of Calculus tells you, if z(t) is continuous then
y(b) - y(a) = int_a^b z(t) dt. Assuming the limit of y(a) as a -> -infinity
is L, that implies y(b) - L = int_{-infinity}^b z(t) dt (that improper
integral being defined as the limit of integrals from a to b as
a -> -infinity). So your formula is right if L happens to be 0, but not
otherwise. If the limit of y(a) as a -> -infinity exist, then the improper
Sorry, I don't know how the word "doesn't" disappeared here.
integral doesn't exist either.
Dr. Israel:
The limit of q(t) is most likely zero as t goes to -inf because it is
a charge, and those are known to decay to nothingness.
Electrical charges decay? That's news to me, not to mention
Maxwell...
Any real electrical circuit is unlikely to have existed from "time
immemorial".
It was set up at some finite time in the past, and anything that your
mathematical model says about it before that time is a fictional
extrapolation. In that context, there's no reason to assume a limit
of 0.
You got me there, because I know next to nothing about physics. All I
know is that after a time, my batteries don't work anymore. So, I
reflected about the vertical axis. This was evidently a bad
application of reflection.
Although I heard Maxwell made very good coffee.
Brian
.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
- Follow-Ups:
- Re: help: first year calculus applied to intro EE
- From: Brian VanPelt
- Re: help: first year calculus applied to intro EE
- References:
- help: first year calculus applied to intro EE
- From: TTroy
- Re: help: first year calculus applied to intro EE
- From: Robert Israel
- Re: help: first year calculus applied to intro EE
- From: Brian VanPelt
- Re: help: first year calculus applied to intro EE
- From: Robert Israel
- help: first year calculus applied to intro EE
- Prev by Date: Re: Vector space query
- Next by Date: Re: How to prove that the space of signed measures is complete?
- Previous by thread: Re: help: first year calculus applied to intro EE
- Next by thread: Re: help: first year calculus applied to intro EE
- Index(es):
Relevant Pages
|
