|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?
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?
No. But you can remember them better if you have an idea how they
|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
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
|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
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?
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
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