TF ErrorAna PropOfErr

A quantity which is calculated using quantities with known uncertainties will have an uncertainty based upon the uncertainty of the quantities used in the calculation.

To determine the uncertainty in a quantity which is a function of other quantities, you can consider the dependence of these quantities in terms of a tayler expansion

Consider a calculation of a Table's Area

$A= L \times W$

The mean that the Area (A) is a function of the Length (L) and the Width (W) of the table.

$A = f(L,W)$

The Taylor series expansion of a function f(x) about the point a is given as

$f(x) = f(a) + f^{\prime}(x)|_{x=a} \frac{x}{1!} + f^{\prime \prime}(x)|_{x=a} \frac{x^2}{2!} + ...$

$= \sum_{n=0}^{infty} f^{(n)}(x)|_{x=a} \frac{x^n}{n!}$

For small values of x (x << 1) we can expand the function about 0 such that

$\sqrt{1+x} = \sqrt{1-0} | \frac{1}{2}(1+x)^{-1/2}|_{x=0} \frac{x^1}{1!}+ \frac{1}{2}\frac{-1}{2}(1+x)^{-3/2}|_{x=0} \frac{x^2}{2!}$

$=1 + \frac{x}{2} - \frac{x^2}{4}$

The talylor expansion of a function with two variable is given by

$f(x_1, x_2)=f({x_o}_1, {x_o}_2)+(x_1-{x_o}_1) \frac{\partial f}{\partial x_1}\bigg |_{(x_1 = x_{01}, x_2 = x_{02})} +(x_2-{x_o}_2) \frac{\partial f}{\partial x_2}\bigg |_{(x_1 = x_{01}, x_2 = x_{02})}$

or

$f(x_1, x_2)-f({x_o}_1, {x_o}_2)=(x_1-{x_o}_1) \frac{\partial f}{\partial x_1}\bigg |_{(x_1 = x_{01}, x_2 = x_{02})} +(x_2-{x_o}_2) \frac{\partial f}{\partial x_2}\bigg |_{(x_1 = x_{01}, x_2 = x_{02})}$

The term

$f(x_1, x_2)-f({x_o}_1, {x_o}_2)$

represents a small fluctuation of the function from its average.

If we ignore higher order terms in the Taylor expansion ( this means the fluctuations are small)

and

Based on the Definition of Varia

\$Delta A = A- A_0 =f(L,W)-f(L_o,W_0)$ = fluctuation of the Area

and simularly

$\Delta L = L-L_0$ and $\Delta W = W-W_0$

Then

$\Delta A = \Delta L \frac{\partial A}{\partial L} \bigg |_{L_0,W_0} + \Delta W \frac{\partial A}{\partial W} \bigg |_{L_0,W_0}$