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Added Flow and Vector Fields

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Christian Risi 2025-11-27 21:50:03 +01:00
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Chapters/15-Appendix-A/INDEX.md
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@ -120,6 +120,66 @@ H(f) = \begin{bmatrix}
\end{bmatrix}
$$
## [Flow](https://en.wikipedia.org/wiki/Flow_(mathematics))[^wiki-flow]
A flow over a set $A$ is a mapping of $R$ over $A$:
$$
a \in A, t \in \R \\
\varphi(a, t) \in A
$$
Moreover, since $\varphi(a, t) \in A$ it also applies:
$$
a \in A, t \in \R, s \in \R \\
\begin{aligned}
\varphi(\varphi(a, t), s) &= \varphi(a, t + s) \in A \\
\varphi(a, 0) &= a
\end{aligned}
$$
In other words, applying a flow over a flow of a variable is like applying
the flow over the variable and the sum of real numbers (think of summing times).
Also, 0 is the neutral element of a flow.
## [Vector Field](https://en.wikipedia.org/wiki/Vector_field)[^wiki-vector-field]
It is a mapping from from a set $A \subset \R^n$ so that:
$$
V: A \rightarrow \R^n
$$
So, this means that for each element of $A$, which we can consider point, it
associates another vector, which we may consider a velocity (but also points).
So, in a way, it can be seen as the amount of movement of that point in space.
## Change of Variables in probability[^stack-change-var]
let's change from 2 random variables, $X$ and $Y$ where $X$ has a CDF that is $F_X$
and $Y = g(X)$ and $g$ is monotonic:
$$
\begin{aligned}
P(Y \leq y) &= P(g(X) \leq y) = P(g^{-1}(g(X)) \leq g^{-1}(y)) = \\
&= P(X \leq x) \rightarrow \\
\rightarrow F_Y(y) &= F_X(x) = F_X(g^{-1}(y))
\end{aligned}
$$
Now, let's derive both handles of the equation for y:
$$
f_Y(y) = f_X(g^{-1}(y)) \cdot \frac{d\, g^{-1}(y)}{d \, y}
$$
> [!NOTE]
> In case x and y are in higher dimensions, the last term is the determinant of
> the Jacobian matrix, or Jacobian
[^khan-1]: [Khan Academy | Laplace Intuition | 9th November 2025](https://www.youtube.com/watch?v=EW08rD-GFh0)
[^wiki-cross-entropy]: [Wikipedia | Cross Entropy | 17th November 2025](https://en.wikipedia.org/wiki/Cross-entropy)
@ -127,3 +187,9 @@ $$
[^wiki-entropy]: [Wikipedia | Entropy | 17th November 2025](https://en.wikipedia.org/wiki/Entropy_(information_theory))
[^wiki-pca]: [Wikipedia | Principal Component Analysis | 18th November 2025](https://en.wikipedia.org/wiki/Principal_component_analysis#Computation_using_the_covariance_method)
[^wiki-flow]: [Wikipedia | Flow (Mathematics) | 23rd November 2025](https://en.wikipedia.org/wiki/Flow_(mathematics))
[^wiki-vector-field]: [Wikipedia | Vector Field |23rd november 2025](https://en.wikipedia.org/wiki/Vector_field)
[^stack-change-var]: [StackExchange | Derivation of change of variables of a probability density function? | 25th November 2025](https://stats.stackexchange.com/questions/239588/derivation-of-change-of-variables-of-a-probability-density-function)