6760, Math 21a, Fall 2009
Frequently asked questions of Math 21a Fall 2008
Course head: Oliver Knill
Office: SciCtr 434
Send questions of potential general interest to math21a@fas.harvard.edu.

Question:On the handout with the list of five PDEs that we should know, do we need to know the constants that are present in each equation (they aren't on the sheet but they are used in the textbook)? For instance, for the advection equation, f_t = (constant)*f_x. In the case of the Burger Equation, for instance, the 'b' that multiplies with f_xx stands for viscosity. Should we know these for all five equations? If so, where would be a good place to look for their definitions? Answer: You should recognize and be able to match them, also if there are constants. You do not have to worry too much about it.
Question:Do we need to know how to derive the 5 PDEs? Answer: No. But you can remember them better if you have an idea how they are derived.
Question:Since we need to know how to recognize the PDEs, should we then know what common types of each PDE look like, i.e., their general solutions? In that case, are the examples on the PDE handout (such as f(x,t) = e^-(x+t)^2 for the Advection Equation) sufficient or are there more types of general solutions (common forms of these 5 PDEs) that we should know? Answer: No. Finding solutions to PDE's is an other course. In Math 21b for example, we learn how to find solutions of the heat or wave equation using Fourier theory.
Question:We used f(x,t) in our PDEs in class (that's also the way that it is in the book) while the handout uses f(t,x). Does that have a physical significance, i.e., doesn't 't' usually come as the second term inside of f? f(x,t) instead of f(t,x)? Which would be a better way to learn the 5 PDEs? Answer: Answer 4) there is no uniform rule how the variables are used. You should be able to recognize it in both cases. Both can be used.
Question:Are Differentials (bottom of page 774 to top of page 776) part of the syllabus? Answer: The term is an old relict: Newton used the name "fluxion", Leibnitz the name "differentials". The second expression has survived until now, but only in calculus textbooks. There is a modern justification of differentials using so called non standard analysis. But this needs some work since it requires an extension of language. One can very do well without differentials: just evaluate the linear function L(x,y) = f(a,b) + fx(a,b) (x-a) + fy(a,b) (y-b) near a given point (a,b) to get an estimate of f(x,y) near (a,b). There is no need to use differentials similarly that one can survive with differences like f(x+h)-f(x) and estimate them using the derivative as f'(x) (x-h). The term still is used by Mathematicians when they talk about differential forms, but these are different beasts and are very precisely defined.
Questions and comments to knill@math.harvard.edu
Math21b | Math 21a | Fall 2009 | Department of Mathematics | Faculty of Art and Sciences | Harvard University