Propagation uncertainties for monomions

Question

Im trying to propagate uncertainties for 1/i where i is the position of image in Gauss equation, my textbook show me this:

For $w=x^m,\text{ } |\frac{\sigma_ w}{w}|=|m\frac{\sigma _x}{x}|$

but, I know that $|\frac{\sigma _w}{w} |= \sqrt{(\frac{\partial w}{\partial x}\sigma _x)^2+(\frac{\partial w}{\partial y}\sigma _y)^2+...}$

So, for $g=\frac{1}{i}, |\frac{\sigma_g}{g}|=\sqrt{(\frac{\partial g}{\partial i}\sigma _i)^2}=\sqrt{(\frac{\sigma _i}{i^2})^2}=|\frac{\sigma _i}{i^2}|=|\frac{g\sigma _i}{i}|$

Thats clearly different from general formula in the textbook. What is wrong here?

Answer 1

Dimensional analysis reveals an algebra error in your last equalities: if $g = 1/i$ is dimensionful, $\frac{\sigma_g}{g}$ is a dimensionless ratio, but $\frac{\sigma_i}{i^2}$ is not.

The error is in your misremembered expression for error propagation using partial derivatives. Use instead

$$ \sigma_w^2 = \left( \frac{\partial w}{\partial x}\sigma_x \right)^2 + \left( \frac{\partial w}{\partial y}\sigma_y \right)^2 + \cdots $$

so that all the terms have the same units as $w^2$, regardless of the units for $x$ and $y$.