Favorites: write these first!
philosophy:
- math as "closure"
- chaitin's nightmare
ped:
- grading
(z-scores, etc.: model of students)
(goal as assessment, or just relative performance)
(t-test!)
linalg
- spectral theorem
(proof, then explanation)
- simplex method
[- note on implicit/explicit being dual]
calc
- implicit/explicit
- exp as (1+x/n)^n and \sum x^n/n!
(solution to diff eqn; also can expand)
- teaching log
(as inverse of exp: use exp_b, log_b notation,
use x and y columns,
use transparent paper and flip)
- integrating x^n
- integrating inverse function
(\int_1^x log = x \log x - x + 1
\int_0^y exp = exp y
)
\int_0^x arctan = x \arctan x - \half \log 1+x^2
\int_0^y tan = \log \sec x = - \log \cos x
yay!
The x f(x) is a give-away!
[NB: \log x = \int 1/x ; \arctan x = \int 1/(1+x^2);
these are the only irreducible/interest rational functions]
...via gluing together into rectangle
[You can do exactly the same thing for inverse hyperbolic functions,
and makes a fun exercise if you must]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Gems:
calc:
- Geometric pictures of many integrals (including $\int x^n$)
- Lagrange multipliers as coordinates of the normal component
- sup \& inf via lattices and lcm \& gcm
- Interpretation of e: (1+x/n)^n = \sum x^n/n!
lin alg:
- Commuting operators and the Nullstellensatz
- Topology of projective spaces and flag manifolds (via Morse functions)
and applications to eigenvectors and linear algebra
POVs:
- The implicit/explicit duality (behind much solving of equations)
- See things from many POVs (e.g., linear algebra via module theory, topology, etc.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Done: