We analyze the local field of stellar tangential velocities for a sample of 42,339 nonbinary Hipparcos stars with accurate parallaxes, using a vector spherical harmonic formalism. We derive simple relations between the parameters of the classical linear model (Ogorodnikov-Milne) of the local systemic field and low-degree terms of the general vector harmonic decomposition. Taking advantage of these relationships, we determine the solar velocity with respect to the local stars of (VX,VY,VZ)=(10.5,18.5,7.3)+/-0.1 km s-1 not corrected for the asymmetric drift with respect to the local standard of rest. If only stars more distant than 100 pc are considered, the peculiar solar motion is (VX,VY,VZ)=(9.9,15.6,6.9)+/-0.2 km s-1. The adverse effects of harmonic leakage, which occurs between the reflex solar motion represented by the three electric vector harmonics in the velocity space and higher degree harmonics in the proper-motion space, are eliminated in our analysis by direct subtraction of the reflex solar velocity in its tangential components for each star. The Oort parameters determined by a straightforward least-squares adjustment in vector spherical harmonics are A=14.0+/-1.4, B=-13.1+/-1.2, K=1.1+/-1.8, and C=-2.9+/-1.4 km s-1 kpc-1. The physical meaning and the implications of these parameters are discussed in the framework of a general linear model of the velocity field. We find a few statistically significant higher degree harmonic terms that do not correspond to any parameters in the classical linear model. One of them, a third-degree electric harmonic, is tentatively explained as the response to a negative linear gradient of rotation velocity with distance from the Galactic plane, which we estimate at ~-20 km s-1 kpc-1. A similar vertical gradient of rotation velocity has been detected for more distant stars representing the thick disk (z>1 kpc), but here we surmise its existence in the thin disk at z<200 pc. The most unexpected and unexplained term within the Ogorodnikov-Milne model is the first-degree magnetic harmonic, representing a rigid rotation of the stellar field about the axis -Y pointing opposite to the direction of rotation. This harmonic comes out with a statistically robust coefficient of 6.2+/-0.9 km s-1 kpc-1 and is also present in the velocity field of more distant stars. The ensuing upward vertical motion of stars in the general direction of the Galactic center and the downward motion in the anticenter direction are opposite to the vector field expected from the stationary Galactic warp model.