Math 21a, Fall 2007
Frequently asked questions of Math 21a Fall 2007
Course head: Oliver Knill
Office: SciCtr 434
Send questions of potential general interest to

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 utt = uxx, 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
 Dv 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 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 0-534-41004-9 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.
Questions and comments to
Math21b | Math 21a | Fall 2007 | Department of Mathematics | Faculty of Art and Sciences | Harvard University