Question:In the checklist distributed at the review, scalar line integrals were mentioned.
Do we have to know about them?

Answer:
No, we look at scalar integrals int f(r(t)) r'(t) dt only in the case
when f=1, that is if we look at the arc length. The only line integrals
we consider are integrals int F(r(t)).r'(t) dt .

Question:I found a discrepancy between the checklists
on the website and my book. The checklist says that the wave equation is
u_{tt} = u_{xx, but the book says that
utt = a2 uxx, where a is a constant that
"depends on the density of the string and on the tension in the string." Is
there a difference? IF so, which one is right? If not, why not?
} 
Answer:
It is the same equation. The version with the constant allows to change the
speed of the wave by changing a parameter. By changing time in
the version with the constant, you get the version without the constant
A physisist usually cares about the constant, a mathematician works in
units in which the constant is 1.

Question:I am confused as to when the flux through a closed
surface is equal to zero. I know that according to the divergence theorem if
Div F is zero then the flux through the closed surface is zero. I also know
that if E=Curl F then the flux through a closed surface over vector field E is
zero according to the divergence theorem. However, I am wondering if these are
the only stipulations. Does a conservative vector field have anything to do
with the flux, or is that irrelevant?

Answer:
You are right that if the divergence is zero (and being the curl of an other vector field is
a special case), then you have a zero flux through any closed
surface. This is if and only if: fields defined in the entire space for which the
flux through all closed surfaces are zero are exactly the divergence free fields.
No, the gadient does not say anything about the flux through a closed
surface. It is the curl which decides whether all line integrals over closed
loops disappear. For fields which are defined in the entire space, the curl is
identically zero if and only if the field is a gradient field.

Question:It seems that some shapes seem to always have a flux of 0.
For instance, in problems I have seen, the flux
through a cube seems to always be zero. Is this always true, or just a
coincidence. If it is not true, can you please provide me with a counter
example?

Answer:
If you take a vector field with a constant divergence for example like F(x,y,z) = (x,0,0),
which has divergence 1. If you integrate the divergence over a solid, you get the
volume of the solid. And this is by the divergence theorem the flux through the boundary
of the solid.

Question:Do you know the date of the final exam yet?

Answer:
We do not know the final exam schedule yet. The official registrars page is
here.

Question:Why do we take directions as unit vectors but allow directional derivatives
to have nonunit vectors?

Answer:
The directional derivative can be defined for all vectors v as
D_{v} f = f . v
But a "direction" is a unit vector v. If a problem does not give you v but asks
for a "direction", then chose a unit vector.
We did not extend the definition of "direction" to arbitrary vectors.
"Directions" are vectors of length 1. For the homework problems:
 11.6: 18 Natural to chose a unit vector but ok to give D_v f with a scaled nonzero vector.
 11.6: 24 Find the 'directions' means 'find a unit vector'. The answer is a unit vector.
 11.6: 26 Because the answer is qualitative, it does not matter.
 11.6: 28 a) any nonzero correct vector ok, b) and c) you look for a direction, a vector of length 1.
 11.6: 30 a),b) assume you walk with speed 1 but any nonzero speed would work
in c) you look for a direction, a vector of length 1

Question:Will there be homework assigned on Wednesday before Thanksgiving?

Answer:
Wednesday before thanksgiving and Monday after thanskgiving are regular
classes. The topic on Wednsday before thanksgiving will be surface integrals.
If you want to plan ahead, there will be homework from Monday to Wednesday
and from Wednesday to Monday after thanksgiving for the MWF sections and
homework from Tuesday to Tuesday for the TTh sections. If you are unable to
do the homework just before thanksgiving starts, you might consider using
one of the three jokers for that day. The homework will be relatively straight
forward so that you can do it on Wednesday before the holiday starts.

Question:Can we use Mathematica for our homework?

Answer:
We encourage to use it but urge you to do the computations
also by hand. A wise strategy is to use Mathematica to check
your work. Note that we do not allow any computers during exams
and homework is an important training for exams. If you use
Mathematica as help, acknowledge that in your homework paper
and print out the notebook, you used similiarly as you use other
sources.
Is this course curved?

Question:Is there class on Friday October 12?

Answer:
Yes, Calculus classes take place as usual.
The inauguration of Drew Faust
takes place in the afternoon and does not interfere with classes.

Question:How do I submit a question to this FAQ list?

Answer:
Just send an email to math21a@fas or knill@math.harvard.edu or ask a section leader. If we
see the question repeated, we will post it.

Question:Which ISBN does the book have?

Answer:
Stewarts Multivariable Calculus Concepts and Context 3: ISBN 0534410049
But be careful.

Question:When will homework be posted?

Answer:
We try to keep 2 weeks HW posted ahead. At least 1 week.

Question:Do we need to buy the solution book?

Answer:
It is handy to have it. Often people share one. The Harvard Coop
should have copies. You can also buy it online.

Question:When will I get my section assignment?

Answer:
It will be sent to you on Friday 5 PM latest by email.

Question:Do I need any programming skills to do the computer algebra project?

Answer:
The Mathematica project can be done without any programming skills. No programming
experiences are necessary. As in the past, it will be a creative assignment,
where you submit an original plot of a surface or other objects. We plan to make
a Mathematica workshop also this semester to get you started.
