Notes on bias and covariance matrix of the angular power spectrum on small sky maps

We compute the effects induced by the use of small CMB maps on the measurement of the $\text{cl}l$ coefficients of the angular power spectrum and show that small systematic effects have to be taken into account. We also compute numerically the cosmic variance and covariance of the $\text{cl}l$ spectrum for various spherical cap like maps. Comparisons with simulations are presented. The calculations are done using the standard method based on the spherical harmonic transform or using the temperature angular correlation spectrum.