port from mathematics-physics notes

docs/physics/mathematical-physics/error-analysis/maximum-error.md

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+# The maximum error
+
+## Determining the transformed maximum error
+
+In this section a method will be postulated and derived under certain assumptions to determine the maximum error, after a transformation with a map $f$. 
+
+> *Definition 1*: let $f: \mathbb{R}^n \to \mathbb{R} :(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y$ be a function that maps independent measurements with a corresponding maximum error to a new quantity $y$ with maximum error $\Delta_y$ for $n \in \mathbb{N}$. 
+
+In assumption that the maximum errors of the independent measurements are small the following may be posed. 
+
+> *Postulate 1*: let $f:(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y$, the maximum error $\Delta_y$ may be given by
+>
+> $$
+>   \Delta_y = \sum_{i=1}^n | \partial_i f(x_1, \dots, x_n) | \Delta_{x_i}, 
+> $$
+>
+> and $y = f(x_1, \dots, x_n)$ correspondingly for all $(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \in \mathbb{R}^n$. 
+
+??? note "*Derivation*:"
+
+    Will be added later.
+
+With this general expression the following properties may be derived.
+
+### Properties
+
+The sum of the independently measured quantities is posed in the following corollary. 
+
+> *Corollary 1*: let $f:(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y$ with $y$ given by
+>
+> $$
+>   y = f(x_1, \dots, x_n) = x_1 + \dots x_n,
+> $$
+>
+> then the maximum error $\Delta_y$ may be given by
+>
+> $$
+>   \Delta_y = \Delta_{x_1} + \dots + \Delta_{x_n},
+> $$
+>
+> for all $(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \in \mathbb{R}^n$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+The multiplication of a constant with the independently measured quantities is posed in the following corollary. 
+
+> *Corollary 2*: let $f:(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \mapsto f(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \overset{.} = y \pm \Delta_y$ with $y$ given by
+>
+> $$
+>   y = f(x_1, \dots, x_n) = \lambda(x_1 + \dots x_n),
+> $$
+>
+> for $\lambda \in \mathbb{R}$ then the maximum error $\Delta_y$ may be given by
+>
+> $$
+>   \Delta_y = |\lambda| (\Delta_{x_1} + \dots + \Delta_{x_n}),
+> $$
+>
+> for all $(x_1 \pm \Delta_{x_1}, \dots, x_n \pm \Delta_{x_n}) \in \mathbb{R}^n$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+The product of two independently measured quantities is posed in the following corollary.
+
+> *Corollary 3*: let $f: (x_1 \pm \Delta_{x_1}, x_2 \pm \Delta_{x_2}) \mapsto f(x_1 \pm \Delta_{x_1}, x_2 \pm \Delta_{x_2}) \overset{.} = y \pm \Delta_y$ with $y$ given by
+>
+> $$
+>   y = f(x_1, x_2) = x_1 x_2,
+> $$
+>
+> then the maximum error $\Delta_y$ may be given by
+>
+> $$
+>   \Delta_y = \frac{\Delta_{x_1}}{|x_1|} + \frac{\Delta_{x_2}}{|x_2|}, 
+> $$    
+>
+> for all $(x_1 \pm \Delta_{x_1}, x_2 \pm \Delta_{x_2}) \in \mathbb{R}^2$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Combining measurements
+
+If by a measurement series $m \in \mathbb{N}$ values $\{y_1 \pm \Delta_{y_1}, \dots, y_m \pm \Delta_{y_m}\}$ have been found for the same quantity then 
+
+$$
+    [y \pm \Delta_y] = \bigcap_{i \in \mathbb{N}[i \leq m]} [y_i \pm \Delta_{y_i}],
+$$
+
+the overlap of all the intervals with $[y \pm \Delta_y]$ denoting the interval in which the real value exists.