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| Question:What kind of partial differential quations can we solve using Fourier?
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Answer:
Any linear equation of the form ut = p(D2)
or utt = p(D2)
where p is a polynomial. In the first case, define
ln = p(-n2). The solution is
u(t,x) = sumn bn exp(lnt) sin(n x)
where bn are the Fourier coefficients of the initial
condition u(0,x). Similar in the wave equation part.
If you want to dig further (we do not cover this in this course),
you can even solve systems of the form
ut = p(D2) + f(t) or
utt = p(D2) + f(t). We solve then the
homogeneous case as before and use a "cookbook" or operator method
find one solution of the inhomogeneous equations for one of the
Fourier modes.
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| Question:When solving for a closed solution to dynamical system questions, what
happens if we cannot form an eigenbasis?
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Answer:
Yes, there is a general method which allows to solve such systems
but it is beyond this course. The theorem to use is called the
Normal form theorem. It allows to find a matrix in which the
matrix A restricted to an eigenspace is a "shear" like in
d/dt x = x + y
d/dt y = y
This is a non diagaonalizable case. To solve it first solve the
second equation y(t) = y(0) et. No solve the first
x' - x = y(0) et which has (use the cookbook
method) the solution x(0) et + t y(0) et.
If you understand this example, you see what is going on in general.
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| Question:In section 2.4 partitioned matrix are used.
Do we have to know about that.
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Answer:
The concept of partitioned matrices allows to do many
things faster. You look at some matrices as matrices
of matrices. For example, for
| 2 3 0 0 | | A 0 |
| 3 5 0 0 | = | 0 B |
| 0 0 1 2 |
| 0 0 3 7 |
you can immediately see the inverse
| A-1 0 | | 5 -3 0 0 |
| 0 B-1 | = | -3 2 0 0 |
| 0 0 7 -2 |
| 0 0 -3 1 |
by inverting the submatrices. The concept is in general useful
only if there are submatrices which are zero. It is a time saver
because inverting a 4x4 matrix with Gauss Jordan is slower
than doing two inversions of 2x2 matrices and for 2x2 matrices,we
can write down the inverse directly: swap the diagonal, negate the
side diagonals and divide by the determinant.
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| Question:For 2.4 number 28, is there a systematic way to solve that or is it
mostly guess and check? Or did I only arrive at an answer because I was
lucky ?
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Answer:
this was a discovery task. There will be a more systematic
way later when we know about other topics where it will be
possible to derive this systematically. This result is important
to keep in mind. Such matrices even have a name, they are called
nilpotent.
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| Question:Will we be allowed to use calculators during the exam?
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Answer:
No. As in Math21a, we do not allow calculators in Math21b.
Consequently, it is a good idea to limit the use of calculators
and computer algebra systems when doing homework. It can be a good
idea to check work using the calculator, but not to do the problems
primarily. You should practice as much as possible to do things by
hand. In the first week, row reduction is an important algoririthm
to master on paper.
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| Question: |
Answer: |
| Question:Which page should I bookmark, the iSites page
or the FAS page?
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Answer:
It is safe to bookmark the FAS page.
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