Mapping the kinematic parameters of the Galaxy from the Gaia EDR3 red giants and sub-giants

We present the results of a kinematic analysis of red giants and subgiants whose centroids are in the plane of our Galaxy. For this, the positions, parallaxes, proper motions, and radial velocities of these stars from the $Gaia$ EDR3 catalog were used. We applied two approaches to obtain kinematic parameters. The first approach -- solving the equations of the Ogorodnikov--Milne model with respect to 12 kinematic parameters -- is generally accepted, but has a number of disadvantages. The second approach developed by us is to find the components of galactocentric centroid's velocity and their partial derivatives with respect to coordinates directly from differential equations for the stellar velocity field. To calculate the kinematic parameters by the methods mentioned above, same stellar samples were used. From these samples spherical regions with a radius of 1 kpc were selected, the centers of which were located strictly in the Galactic mid-plane at the nodes of the coordinate grid ($x_{gal}$, $y_{gal}$) of a rectangular Galactocentric coordinate system with step 100 pc. The region of the Galaxy under study occupies the coordinate interval $115^\circ < \theta < 245^\circ$, 0 kpc$ < R < $16 kpc, -1 kpc$ < z < $1 kpc. We show the behavior of local kinematic parameters as well as global parameters such as the circular velocity of stars as a function of galactocentric coordinates. For the first time, the components of centroids' spatial velocities and all their partial derivatives as well as their behavior as a function of galactic coordinates have been derived. The behavior of the $\partial V_R/\partial \theta$ and $\partial V_\theta/\partial \theta$ parameters as a function of galactic coordinates has been derived for the first time.