Tweet
Login
Mathematics Crystal
You may switch between
tex
and
pdf
by changing the end of the URL.
Home
About Us
Materials
Site Map
Questions and Answers
Skills
Topic Notes
HSC
Integration
Others
Tangent
UBC
UNSW
Calculus Advanced
Challenges
Complex Numbers
Conics
Differentiation
Integration
Linear Algebra
Mathematical Induction
Motion
Others
Polynomial Functions
Probability
Sequences and Series
Trigonometry
/
Topics /
Calculus Advanced /
Combined Uncertainty.tex
--Quick Links--
The Number Empire
Wolfram Mathematica online integrator
FooPlot
Calc Matthen
Walter Zorn
Quick Math
Lists of integrals
List of integrals of trigonometric functions
PDF
\documentclass[10pt]{article} \usepackage{amssymb,amsmath} \usepackage[hmargin=1cm,vmargin=0.5cm]{geometry} \begin{document} {\large Combined Uncertainty} \begin{align*} \text{\bf Basics:}\quad& \text{Uncertainty is how much doubt about a measurement. e.g. If there is 95\% of confidence that the length of}\\ &\text{a rod is between 9.95 m and 10.05 m, we say the uncertainty is 0.05 m, i.e. 10.00 $\pm$ 0.05 m.}\\ \\ &\text{\it Combined Uncertainty \rm is the uncertainty of a value computed from other values or measurements.}\\ &\text{The formula to compute the Combined Uncertainty is}\\ \\ &\Delta z=\sqrt{\sum_{k=1}^n\left(\Delta x_k\cdot\frac{\partial f}{\partial x_k}\right)^2}\qquad\text{where} \quad z=f(x_1,x_2,\cdots,x_n).\qquad\text{(Uncertainty is always positive.)}\\ \\ &\text{This means that if the value of $z$ is computed from other values or measurements $x_1,x_2,\cdots,x_n$ through}\\ &\text{function $f()$, then $\Delta z$, the uncertainty of $z$, can be computed through the above formula.}\\ \\ &\text{Sometimes the function $f()$ is implied:}\quad\boxed{\Delta z=\sqrt{\left(\Delta x_1\cdot\frac{\partial z}{\partial x_1}\right)^2+\left(\Delta x_2\cdot\frac{\partial z}{\partial x_2}\right)^2+\cdots+\left(\Delta x_n\cdot\frac{\partial z}{\partial x_n}\right)^2}\;.}\\ \\ \text{\bf Addition:}\quad&z=\sum_{k=1}^n x_k=x_1+x_2+\cdots+x_n\;.\\ &\because\frac{\partial z}{\partial x_k}=1\;,\qquad \therefore\Delta z=\sqrt{\sum_{k=1}^n\big(\Delta x_k\big)^2}\quad. \qquad\text{i.e. }\boxed{\Delta z=\sqrt{\Delta x_1^2+\Delta x_2^2+\cdots+\Delta x_n^2}\;.}\\ &\text{Subtraction means $-x_k$ is among the terms, so $\frac{\partial z}{\partial x_k}=-1$ and $\left(\frac{\partial z}{\partial x_k}\right)^2=1$}\text{; the same formula will apply.}\\ \\ \text{\bf Index:}\quad&z=x^n\;.\\ &\because\frac{\partial z}{\partial x}=nx^{n-1}=\frac{nz}{x}\;,\qquad \therefore\Delta z=\sqrt{\left(\Delta x\cdot\frac{nz}{x}\right)^2}=\left|\frac{n\Delta x}{x}\cdot z\right|\quad.\\ &\text{Divide both sides by $|z|$ :}\quad\boxed{\left|\frac{\Delta z}{z}\right|=\left|\frac{n\Delta x}{x}\right|\;.}\\ \\ \text{\bf Multiplication:}\quad&z=\prod_{k=1}^n x_k=x_1\cdot x_2\cdot\cdots\cdot x_n\;.\\ &\because\frac{\partial z}{\partial x_k}=\prod_{r=1,r\neq k}^n x_r\;\;=\frac{z}{x_k}\;,\qquad \therefore\Delta z=\sqrt{\sum_{k=1}^n\left(\Delta x_k\cdot\frac{z}{x_k}\right)^2}\quad.\\ &\text{Divide both sides by $|z|$ :}\quad\boxed{\frac{\Delta z}{|z|}=\sqrt{\left(\frac{\Delta x_1}{x_1}\right)^2+\left(\frac{\Delta x_2}{x_2}\right)^2+\cdots+\left(\frac{\Delta x_n}{x_n}\right)^2}\;.}\\ &\text{Division can be taken as $x^{-1}$ and $\left(\frac{-1\cdot\Delta x}{x}\right)^2=\left(\frac{\Delta x}{x}\right)^2$} \text{; the same formula will apply.}\\ \\ &\text{Generally, if }z=\prod_{k=1}^n x_k^{a_k}=x_1^{a_1}\cdot x_2^{a_2}\cdot\cdots\cdot x_n^{a_n}\;,\quad \frac{\Delta z}{|z|}=\sqrt{\left(\frac{a_1\Delta x_1}{x_1}\right)^2+\left(\frac{a_2\Delta x_2}{x_2}\right)^2+\cdots+\left(\frac{a_n\Delta x_n}{x_n}\right)^2}\;. \end{align*} \end{document}