[BUG] The Hill in a multivariate course

[Smith, Alexander J.]

As you indicate, I put The Hill right after the introduction to
functions of several variables. It then served as something to refer
back to when gradients and directional derivatives were introduced.

I think most students got lots out of it. The majority of groups would
express y as a linear function of x and then diff h(x,y(x)) wrt x, and
think this was the slope. This just gives "vertical feet per eastward
mile" but it is a good start. I just decided to not stress out about
this. You get to revisit their mistake over and over when the chain rule
is thrown in, and when directional derivatives are introduced. 

 I remember groups avidly talking about the steepness question (#3). At
first they all started knocking the problem because it seemed the hill
was unrealistically steep, but then realized the units were feet per
mile. This was good, even if they only found dh/dx instead of dh/ds.

Splitting The Valley worked fine for me. There is the abyss of the
chapter on multiple integrals between Valley I and II. When we finally
did get to line integrals, it was fun to have a paper record with the
"distant" past. In Part I, I only had them compute grad(h).dr. This is
3(a) and (b) with a covering up of the integral. Then when you return in
Part II after line integrals are introduced, everybody is confused but
spooked when they find out that back in Part I, they knew what they were
doing. 


Alex

-----Original Message-----
From: bug-bounces at science.oregonstate.edu
[mailto:bug-bounces at science.oregonstate.edu] On Behalf Of Martin Jackson
Sent: Wednesday, January 14, 2004 11:00 AM
To: bug at science.oregonstate.edu
Subject: [BUG] The Hill in a multivariate course

All,

I am preparing to make full use of the Bridge materials in my 
multivariate calculus course that starts next week.  This is a 
semester course (15 weeks) that covers vectors, vector-valued 
functions, derivatives of functions of several variables (partial 
der, gradients, etc), multiple integrals, and vector calculus.  We 
use Strauss/Bradley/Smith (3rd edition, Strauss added as an author to 
Bradley/Smith).

When  I look to lay out Bridge activities through the semester, I see 
a big gap between (Which Way is North?/Acceleration/Finding dr) and 
The Hill if I wait to do The Hill until gradients are covered.  I'm 
considering doing The Hill just after introducing functions of 
several variables (graphs,level curves).  Students would confront the 
ideas of slope and "direction of steepest ascent" before developing 
the conventional computational tools.  This will make all of the 
activity essentially open-ended.  I might revisit the activity after 
we have worked with gradients.

In looking at the schedule of activitites that Alex Smith shared with 
us last fall (his message of August 24, 2003), I see he put The Hill 
in a spot similar to what I intend.  I also note that Alex has The 
Valley split into two parts.  Alex, can you give us a bit more detail 
on this?

Alex also mentioned a gap in the Bridge activities when doing 
multiple integrals.  In previous semesters, I have had students do 
small-group activities to get the volume elements in cylindrical and 
spherical coordinates using geometric reasoning.

I would appreciate any thoughts on this use of The Hill.

Martin
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