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