In the stability analysis of power systems it is extremely important to determine the stability region of a working (equilibrium) point. This paper first gives a quadratic approximation to the boundary sub-manifold of the stability region, which assures the error be of O(||x||/sub 3/). Under certain non-singularity assumption, a precise expression of the submanifold is obtained as a Taylor expansion. The formula is then extended for differential-algebraic systems. Its application to power systems is illustrated via an example. The computation is based on the left semi-tensor product of matrices, which was proposed in [D. Cheng, 2001].