Miscellaneous

Gaussian

From a generic form to a PDF whose CDF approaches 1

$$f(x)=a e^{-\frac{(x-b)^{2}}{2 c^{2}}} \rightarrow g(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}$$

Quick Proof of Bayes Theorem

Partition the possibility space into four sections much like Punnet squares.

$$P(A\cap B) = P(A) P(B | A) = P(B) P(A| B)$$

Solve for the conditional probability

The null space is the set of vectors which makes the output homogenous. See 3B1B for more.

Multivariate calculus is the extension of of univariate calculus, which studies mappings $f: \mathcal{R} \rightarrow \mathcal{R}$ to more generic $\boldsymbol{g}: \mathcal{R}^{n} \rightarrow \mathcal{R}^{m}$; relating a vector, $\boldsymbol{x}$ of some length$n$ to some other vector $\boldsymbol{y}$of length $m$by multiplication with some some rectangular matrix of dimensions, $m \times n$.

• $\boldsymbol{x}$ ... initial configuration

• $\boldsymbol{y}$ ... deformed configuration

• $\boldsymbol{u} = \boldsymbol{y} - \boldsymbol{x}$... motion

For the generic example, the initial configuration is some smooth closed volume in $\mathcal{R}^{3}$; represented as a collection of position vectors, $\boldsymbol{r}$. Equivalent representation is by summation of unit vectors in each of the cartesian directions:

$$\boldsymbol{r}=\sum_{i=1}^{3} x_{i} \boldsymbol{e}_{i} = \overbrace{x_{1} \boldsymbol{e}_{1} + x_{2} \boldsymbol{e}_{2} + x_{3} \boldsymbol{e}_{3} = x_{i} \boldsymbol{e}_{i} }^{\text{ in indicial notation}}.$$

Then in the $\mathcal{R}^{3}$ case, we see:

$\overbrace{\begin{bmatrix} y_1\\ y_2\\ y_3 \end{bmatrix} = \begin{bmatrix} 1+ \frac{u_1}{x_1}&0&0\\ 0& 1+ \frac{u_2}{x_2}&0\\ 0&0& 1+ \frac{u_3}{x_3} \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} = \begin{bmatrix} x_1+ u_1\\ x_2+u_2\\ x_3 +u_3\end{bmatrix}}^{\text{solid mechanics example, see the identity matrix?}}.$