Wiki - Roberto Olivares | Knowledge / Math

Tex Symbols:

http://www.math.union.edu/~dpvc/jsMath/symbols/welcome.html

Integrals:

Integrals via Mathematica Online

Exponentials

e is a very mysterious number.

Linear Algebra

Determinant

\left|\matrix{a_1 & a_2 \\ b_1 & b_2}\right| = a_1 b_2 - a_2 b_1

Inner Product (or pointwise product):

A \cdot B = \left[\matrix{a_1 \\ a_2 \\ a_3}\right] \cdot \left[\matrix{b_1 \\ b_2 \\ b_3}\right] = a_1 b_1 + a_2 b_2 + \ldots a_n b_n = ||a|| \, ||b|| \cos \theta

Outer Product (or matrix product)

A B = \left[\matrix{ \ldots & r_1 & \ldots \\ \ldots & r_2 & \ldots \\ \ldots & r_3 & \ldots \\}\right] \left[\matrix{ \vdots & \vdots & \vdots \\ c_1 & c_2 & c_3 \\ \vdots & \vdots & \vdots }\right] = \left[\matrix{r_1 \cdot c_1 & r_1 \cdot c_2 & r_1 \cdot c_3 \\ r_2 \cdot c_1 & r_2 \cdot c_2 & r_2 \cdot c_3 \\ r_3 \cdot c_1 & r_3 \cdot c_2 & r_3 \cdot c_3 }\right]

Each element in the final matrix is the pointwise product of a row in the A matrix and a column in the B matrix. Which row and column correspond to the elements own r,c coordinate.
Cross Product (note the 23, 13, 12 sequence)

A \times B = \left[\matrix{a_1 \\ a_2 \\ a_3}\right] \times \left[\matrix{b_1 \\ b_2 \\ b_3}\right] = \left|\matrix{a_2 & a_3 \\ b_2 & b_3}\right| \hat{i} - \left|\matrix{a_1 & a_3 \\ b_1 & b_3}\right| \hat{j} + \left|\matrix{a_1 & a_2 \\ b_1 & b_2}\right| \hat{k}

\mathbf{a} \times \mathbf{b} = \mathbf{A}_{\times} \mathbf{b} = \begin{bmatrix}0&-a_3&a_2\\a_3&0&-a_1\\-a_2&a_1&0\end{bmatrix}\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}

Exponents

x^(a*b) = (x^a)^b
x^(a+b) = (x^a)(x^b)
x^(a-b) = (x^a)/(x^b)
x^(-a) = 1/(x^a)

Logarithms & Exponentials

“Log A of X” means “A to what power yields X?”
“ln X” means “e to what power yields X?”

Logarithms

“Log A of X” means “A to what power yields X?”
log (base a) of x = b such that a^b = x
Log10 of 10 = 1 and Log10 of 100 = 2 because (10^2) = 100

Log A of A = 1

Natural Logarithm

“ln a” means “e to what power is a?”

Information Theory:

Tutorial on Entropy and Mutual Information