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

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

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

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

$f(x_1, x_2)=f({x_o}_1, {x_o}_2)+(x_1-{x_o}_1) \frac{\partial f}{\partial x_1}|_{x_1=x_o_1 ,x_2=x_o_2}}+(x_2-{x_o}_2) \left \| \frac{\partial f}{\partial x_2 \right |_{(x_1=x_o_1 , x_2=x_o_2})}$

$f(x_1, x_2) – f({x_o}_1, {x_o}_2) = (x_1-{x_o}_1) \frac{\partial f}{\partial x_1}\Big\lvert_{\substack {x_1={x_o}_1 \\ x_2={x_o}_2}}+(x_2-{x_o}_2) \frac{\partial f}{\partial x_2}\Big\lvert_{\substack {x_1={x_o}_1 x_2={x_o}_2}}$