[BUG] A geometric approach to gradients

[Martin Jackson]

>  >>>>> Martin Jackson writes:
>
>     MJ> Tevian, can you give us more detail on how you want students to
>     MJ> think about your RESOLUTION?  In particular, how do you want them to
>     MJ> think about counting "the number of contour lines" in computing the
>     MJ> number per unit distance?  Does Delta f enter into this explicitly?
>
>Yes, I think it must.  I like talking about the "number" of contour lines
>because I think it's easier for students to comprehend -- you can easily see
>how many lines are crossed, and it's then quite intuitive to see directional
>derivatives as projections, that is, dot products.  I called it "resolution"
>because the natural units are something like "lines per inch".  But of course
>what's really being described is the change in the value of the function.
>
>Be aware that I made this argument initially for a linear function, for which
>I didn't have to distinguish between the graph and the tangent plane.  Any
>effort to make this argument for a nonlinear function should really address
>the fact that the "contours" on the tangent plane only approximate the change
>in the function.  I didn't address this, and felt that most students weren't
>bothered by it.  Not sure if this is your point, nor how careful one must be
>here.
>
>Tevian
>_______________________________________________

In posing my question, I was curious about how you are setting up the 
contour diagrams (for linear functions) in which you are doing the 
counting.  Are you always using Delta z=1 in these?  (In what 
follows, I'll write Dz for Delta z.)

(A small aside: You are probably using a constant value for Dz.  This 
is implicit in the way we generally ask students to think about 
contour diagrams and gradients.  Watching my physics colleague lead 
our integrated calc/phys class the other day, I was struck by the 
need to be careful about this.  He was discussing equipotentials for 
the electric potential U due to a point charge.  The function is 
radially symmetric and goes like 1/r.  In drawing cross-sections of 
the equipotentials, he started with a circle at some arbitrary radius 
and labeled it with an arbitrary potential U1.  He then asked 
students about the equipotentials for 1/2*U1, 1/4*U1 and so on.  The 
resulting diagram does not have a constant value for DU between 
adjacent contours.  Care must be used in thinking about a "contour 
density" in this context.)

Getting back to Tevian's proposal, I was inspired to give this 
approach a try; the timing is right in our calc/phys course as I am 
just introducing derivative for functions of more than one variable. 
Here's a report on the beginning of this experiment.

In yesterday's class, I started with a brief reminder on slope for 
lines in a plane, emphasizing slope as both a ratio Dy/Dx (that is 
independent of position on the line) and as a rate of change in one 
variable with respect to the other. I then turned to planes in space. 
I started by looking at the contour diagram for a generic linear 
function.  In response to my questions, students volunteered that the 
level curves are parallel lines with equal spacing (for constant Dz). 
I drew a point P, a level curve labeled z0 through the point, and 
parallel level curves for z0-Dz and z0+Dz.  I then asked about 
extending the idea of slope.  Students came up with going from P to 
the z0+Dz level curve in the direction parallel to the x-axis, 
measuring Dx and forming the ratio Dz/Dx  and the analogous idea for 
Dz/Dy.  I suggested measuring the distance between the level curves, 
which I labeled Ds, and forming the ratio Dz/Ds.  I then defined a 
vector m as having direction perpendicular to the level curves and 
magnitude equal to Dz/Ds, i.e., the rate of change in z with respect 
to change in the direction of m.   This took about 10 minutes.

Following this, I assigned the small group task of determining the 
vector m for a specific linear function, z=5x+3y+7.  All of the 
groups eventually settled on the following process: Draw the level 
curves for z=0 and z=1.  Pick a point on the z=0 level curve (most 
used the y-intercept).  Find the equation of the line perpendicular 
to the z=0 level curve using the negative reciprocal of that line's 
slope.  Find the point of intersection between the perpendicular line 
and the z=1 level curve.  Compute the distance between the two points 
to get Ds and then get the magnitude of m as 1/Ds.

In a wrap-up discussion, I reminded students how we generalize slope 
to curves in the plane (by zooming in at a point which corresponds to 
taking a limit) and briefly discussed how we will generalize the 
vector m to surfaces in space.

Some observations:
1.  I think students initially suggested Dz/Dx and Dz/Dy because we 
use change in coordinate directions for defining the slope of a line 
in the plane.  I probably also induced this by including a coordinate 
system on my contour diagram when I really did not need one.  In any 
case, I used the student's suggestions as motivation for partial 
derivatives.

2.  In their own approach, students chose Dz=1 for themselves.  In 
follow-up discussion, I suggested an alternate approach of going out 
the perpendicular line any convenient distance (over 5, up 3 is easy 
for this example knowing the relevant slope is 3/5) and then 
determining the value of z at that point to compute Dz and Ds.

3.  I've defined gradient in the linear case to have direction 
perpendicular to the level curves.  Tevian defines the direction to 
be "that in which f increases the fastest."  I'm not sure if one 
approach is significantly better than the other.  For me, the 
direction that is geometrically obvious in looking at the contour 
diagram is the one perpendicular to the level curves so this seems 
like the more natural definition.  The fact that this direction gives 
the greatest rate of change requires a bit of reasoning and seems 
more natural as a conclusion.

4.  In giving the specific example z=5x+3y+7, I used coordinates.  Is 
there some better initial exercise that doesn't use coordinates to 
specify a linear function?