Initial observations of the dayside magnetopause thickness using UCLA magnetometers on board Isee 1 (International Sun-Earth Explorer) and Isee 2 indicate that this boundary may only be a few ion gyroradii thick during periods of observed southward interplanetary magnetic field (IMF). Such a thin current layer can destabilize the collisionless tearing mode, a probable first step in the reconnection sequence. We have calculated the linear growth rate assuming typical magnetopause parameters and find that it is a sensitive function of IMF orientation, magnetopause thickness, and electron number density. We also show that the finite size of the dayside magnetopause necessitates the existence of a two-dimensional tearing mode wave vector spectrum, a consequence of the inability of a thermal electron to maintain Laudau resonance with the wave for a growth period. Implications regarding reconnection are discussed.