> For the complete documentation index, see [llms.txt](https://nkintc.gitbook.io/brainless/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://nkintc.gitbook.io/brainless/steam/mathematics/untitled/miscellaneous.md).

# 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](https://www.youtube.com/watch?v=uQhTuRlWMxw). 

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:

## $$\overbrace{\begin{bmatrix} y\_1\ y\_2\ y\_3 \end{bmatrix} = \begin{bmatrix} 1+ \frac{u\_1}{x\_1}&0&0\ 0& 1+ \frac{u\_2}{x\_2}&0\ 0&0& 1+ \frac{u\_3}{x\_3} \end{bmatrix}  \begin{bmatrix} x\_1\ x\_2\ x\_3 \end{bmatrix} =  \begin{bmatrix} x\_1+ u\_1\ x\_2+u\_2\ x\_3 +u\_3\end{bmatrix}}^{\text{solid mechanics example, see the identity matrix?}}.$$ 


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