Chapter 2 Math typesetting

2.1 Math

 is the best way to typeset mathematics. Donald Knuth designed  when he got frustrated at how long it was taking the typesetters to finish his book, which contained a lot of mathematics. One nice feature of R Markdown is its ability to read LaTeX code directly.

If you are doing a thesis that will involve lots of math, you will want to read the following section.

\[\sum_{j=1}^n (\delta\theta_j)^2 \leq {{\beta_i^2}\over{\delta_i^2 + \rho_i^2}} \left[ 2\rho_i^2 + {\delta_i^2\beta_i^2\over{\delta_i^2 + \rho_i^2}} \right] \equiv \omega_i^2 \]

From Informational Dynamics, we have the following (Dave Braden):

After n such encounters the posterior density for \(\theta\) is

\[ \pi(\theta|X_1< y_1,\dots,X_n<y_n) \varpropto \pi(\theta) \prod_{i=1}^n\int_{-\infty}^{y_i} \exp\left(-{(x-\theta)^2\over{2\sigma^2}}\right)\ dx \]

Another equation:

\[\det\left|\,\begin{matrix}% c_0&c_1\hfill&c_2\hfill&\ldots&c_n\hfill\cr c_1&c_2\hfill&c_3\hfill&\ldots&c_{n+1}\hfill\cr c_2&c_3\hfill&c_4\hfill&\ldots&c_{n+2}\hfill\cr \,\vdots\hfill&\,\vdots\hfill& \,\vdots\hfill&&\,\vdots\hfill\cr c_n&c_{n+1}\hfill&c_{n+2}\hfill&\ldots&c_{2n}\hfill\cr \end{matrix}\right|>0\]

Lapidus and Pindar, Numerical Solution of Partial Differential Equations in Science and Engineering. Page 54

\[ \int_t\left\{\sum_{j=1}^3 T_j \left({d\phi_j\over dt}+k\phi_j\right)-kT_e\right\}w_i(t)\ dt=0, \qquad\quad i=1,2,3. \]

L&P Galerkin method weighting functions. Page 55

\[ \sum_{j=1}^3 T_j\int_0^1\left\{{d\phi_j\over dt} + k\phi_j\right\} \phi_i\ dt = \int_{0}^1k\,T_e\phi_idt, \qquad i=1,2,3 \]

Another L&P (p145)

\[ \int_{-1}^1\!\int_{-1}^1\!\int_{-1}^1 f\big(\xi,\eta,\zeta\big) = \sum_{k=1}^n\sum_{j=1}^n\sum_{i=1}^n w_i w_j w_k f\big( \xi,\eta,\zeta\big). \]

Another L&P (p126)

\[ \int_{A_e} (\,\cdot\,) dx dy = \int_{-1}^1\!\int_{-1}^1 (\,\cdot\,) \det[J] d\xi d\eta. \]