common derivations

1. time dependent from time independent variable:

   \[\forall t: x(t,:) = x(:)\]

2. total uncertainty from random and systematic uncertainty:

   symbol	description	unit	variable name
   \(\mu_{x}\)	total uncertainty for a variable x	same as that of \(x\)	<variable>_uncertainty {:}
   \(\mu^{r}_{x}\)	uncertainty due to random effects for a variable x	same as that of \(x\)	<variable>_uncertainty_random {:}
   \(\mu^{s}_{x}\)	uncertainty due to systematic effects for a variable x	same as that of \(x\)	<variable>_uncertainty_systematic {:}

   The pattern : for the first dimensions can represent any combination of dimensions for which x {:} exists.

   \[\mu_{x} = \sqrt{{\mu^{r}_{x}}^2 + {\mu^{s}_{x}}^2}\]

3. total uncertainty from vertical covariance:

   symbol	description	unit	variable name
   \(S_{x}(i,j)\)	covariance for a variable x	square of that of \(x\)	<variable>_covariance {:,vertical,vertical}
   \(\mu_{x}(i)\)	total uncertainty for a variable x	same as that of \(x\)	<variable>_uncertainty {:,vertical}

   The pattern : for the dimensions can represent any combination of dimensions for which x {:,vertical} exists.

   \[\forall i: \mu_{x}(i) = \sqrt{S(i,i)}\]