[BUG] update

[Tevian Dray]

Here are some belated comments on Stuart's update:

> 2. Finding dr(vector): 
> 
> a.  Since I didn't use the Acceleration lab, I added problem zero from
> Acceleration to this lab.  When trying to give advice to the class as a
> whole while they were working on this lab, the abundance of "r"s made this
> tough (\vec r, \hat r, r the scalar, dr the scalar, dr the vector, etc).

I agree that this is confusing, but don't know a better alternative.  Of the 5
you list, the middle 3 are derived from the coordinate r and must therefore
all use the same base letter, but a different letter, such as "ell", could be
used for the position vector and vector differential, your 1st and 5th items.

> b. Many of my students didn't sketch r hat and theta hat with their
> tails on the unit circle (many had their tips end on the circle).  Maybe
> this should be made explicit in the directions?

Rereading your first comment, I suspect that this question is really about
problem zero on the Acceleration lab.  That problem includes a useful but
unlabeled figure showing rhat and thetahat, which students should be
encouraged to use, and which might resolve this issue.

There is nothing logically wrong with the (nonstandard) convention that
vectors "live" at their tips, rather than at their tails.  But I doubt that's
what these students were thinking.  Rather, I would guess that they wanted all
vectors to live at the origin.  I agree that it is important for students to
realize that this is wrong (and in any case rhat not defined at the origin),
but am not convinced that this should be part of the directions.  Perhaps a
more open-ended question, such as asking where the various vectors live.  Or
perhaps this would be most effective as part of the wrapup.  Comments?

> c.  When they were asked to plot all four vectors (i,j,r hat, theta
> hat) they again, didn't always put the tails at the same point.  Thus, I
> don't think they really saw how all four vectors were related.

Students do need to feel comfortable drawing these vectors, and part of that
is knowing where to put them.  I would again suggest making this part of the
wrapup (and adding an appropriate discussion to the Instructor's Guide).

> 3. The Hill:

>  a.  Many of my students had a hard time drawing a simple picture
> representing the steepness of the hill.  Not really sure why...

It's an open-ended question...

>  b. For the last problem, determining the correct k component can be
> tricky (norm of grad or the norm squared).  If they realized that this
> vector should be tangent to the hill and they recall how to find normal
> vectors to surfaces using gradients, they could quickly check their
> work.  I had one student do this (actually he found the k component by
> making the corresponding dot product equal to zero).

This method is discussed briefly in the Instructor's Guide as enrichment.

> On a related note: In the instructor's guide the directional derivative
> is derived from the master formula as a rate of change with respect to
> distance traveled (the puddle problem).  However, in the hill problem to
> correctly interpret the "steepness" of the hill, you need to realize
> that a directional derivative give the rate of change as a function of
> distance traveled IN THE TOPO MAP (i.e. horizontal distance traveled). 
> Right?  

Right.  Both this comment and the previous one involve possible confusion
between the 2-dimensional topo map and the 3-dimensional hill.  Since the
notion of gradient is in fact only rarely used in this confusing context,
we continue to wonder whether this lab possibly does more harm than good.
(We used to ask explicitly for the normal vector to the hill...)
On the other hand, students seem to like it, and seem to understand the
gradient better for having done it.

Tevian