## Archive for March 2009

### Tech Conference Posters

13 March 2009

I finished two posters in LaTeX for the conference, that took most of a week, about 24 hours.  They came out quite well though, and I’m proud of them.

### Limits

10 March 2009

Tonight, I was reading about the fundamental definition of limits of functions, and for the first time, really understood it.  The context was a limit in multiple variables, as

$\displaystyle\lim_{(x,y)\to (0,0)}f(x,y) = L$

The interesting issue is that in two dimensions like this, showing the limit exists is harder than one dimensions, since there are infinitely many ways to approach (a,b).   The definition for continuity becomes, $f(x,y)$ is continuous at the point $(a,b)$ if the limit exists and is $f(a,b)$, or

$\displaystyle\lim_{(x,y)\to (a,b)}f(x,y) = f(a,b)$

Somehow, I never fully understood the 1D case until I saw the explaination and example in 2D.

### Stuck “easily showing that…” in Modinos

8 March 2009

I’m currently stuck on Modinos’ derivation of the the number of electrons when cross a unit area per unit time, with total energy between $E$ and $E+dE$ and normal energy between $W$ and $W+dW$.

$N(E,W) dE dW = \frac{m}{2 \pi^2 \hbar^3} n(E) dE dW$

where $n(E)$ is the Fermi-Dirac distribution.   The blocking issue is integrating over an unusual region or k-vector 3 space where {E,W} under the triple integral indicates only states with total energy between $E$ and $E+dE$ and normal energy between $W$ and $W+dW$ corresponding to $v_z>0$ included in the integration.

### xyz density functions

7 March 2009

Read through Eisberg & Resnick on the derivation fo the electron gas energy distribution of conduction electrons in a metal.  This book is really good, I’m glad I have it.

$n(E)N(E)dE = \frac{8 \pi V (2m^3)^{1/2}}{h^3} \frac{E^{1/2} dE}{e^{(E-E_F)/kT}+1}$

Now I also finally got clear on “density functions”, the meaning of the dE in a formula, and why they live to be integrated.

### BibTeX references

7 March 2009

I found there is an easy way to get BibTex references from Google Scholar (check out the preferences), now I don’t need to worry about getting some reference management software for BibTeX.   I can keep a .bib file updated easily enough with this.   There is also quite a nice wikibook on LaTeX with a bibliography section.

### Video lectures

6 March 2009

So there are about 33 lectures in the differential math at MIT video series.  There are other useful series too.  I see there is one on Multivariable Calculus, Linear Algebra, and also available is a list of all video lectures.

### MIT 1802 – lecture 3 (Multivariable calculus)

6 March 2009

This lecture was the first I took notes for from the MIT video series on multivariable calculus.  It covered calculating the inverse of a 3×3 matrix.  The answer is, to invert matrix $A$, find the determinant of $A$ and the adjoint of $A$.

$A^{-1} = \frac{1}{det(A)} adj(A)$

The steps are:

1) find the MINORS (a 3×3 matrix)

2) multiply by the COFACTORS (a +/- matrix alternating with a + in upper left corner)

3) TRANSPOSE the matrix (switch rows and columns)

4) MULTIPLY by one over the determinant of A