# 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.

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