[BUG] Re: BUG Digest, Vol 3, Issue 6

[Philip Yasskin]

In reply to Tevian's request, here are two problems I have given on an exam:

1. Duke Skywater is flying the Millenium Eagle through a polaron
field.  His galactic coordinates are   (2300,4200,1600)   measured in
lightseconds and his velocity is   v=(.2,.3,.4)   measured in
lightseconds per second.  He measures the strength of the polaron
field is   p=274 milliwookies   and its gradient is   del p=(3,2,2)
milliwookies per lightsecond.  Assuming a linear approximation for the
polaron field and that his velocity is constant, how many seconds will
Duke need to wait until the polaron field has grown to  
286milliwookies?

2. Duke Skywater is flying the Millennium Eagle through the galactic
hyperon field.  At t=20, Duke's position is r=(20,10,30) lightyears
and his velocity is v=(.1,.3,.2) ((lightyears)/(year)).  At that time,
he measures the hyperon field and its Jacobian to be

	H=( "A COLUMN OF 3 NUMBERS" )Hans
	DH=( "A 3x3 MATRIX OF NUMBERS" )((Hans)/(lightyear))
	
a.What does he measure as the time rate of change of the hyperon field, dH/dt?
b.Using a linear approximation, what would he expect the hyperon field
to be at t=22?

Phil

> ---------- Forwarded message ----------
> From: Tevian Dray <tevian at math.oregonstate.edu>
> To: Bridge Users Group <bug at science.oregonstate.edu>
> Date: Sun, 03 Apr 2005 22:10:02 -0700
> Subject: [BUG] multivariable chain rule
> Can anyone provide a good example of the use of the multivariable chain rule?
> Seems to me that most purported examples don't actually involve the chain rule
> at all.  For instance, if V=IR and 1/R=1/R1+1/R2, the computation of dV/dt
> doesn't really use the chain rule (with V=V(I,R) and everything else a
> function of t), but rather the product and power rules.  In fact, I claim the
> only time one actually needs the multivariable chain rule is when one does NOT
> explicitly know the functional dependence.  In the above example, that means
> not knowing how V depends on I and R.  This is very relevant for theoretical
> discussions, such as changing variables in a PDE.  But what purpose does it
> serve in a multivariable calculus course?
>
> Tevian
--
Philip B. Yasskin
Department of Mathematics
Texas A&M University
3368 TAMU
College Station, TX 77843-3368

yasskin at math.tamu.edu
http://www.math.tamu.edu/~yasskin/

Room:       BLOC 620 I
Office:     979-845-3734
Department: 979-845-3261
FAX:        979-862-4190