A Theorem on Homotopy Paths

We consider the set of points x R n +1 satisfying H ( x ) = 0, where H : R n +1 R n is a C 1 function and 0 is a regular value. This set, H –1 (0), is a C 1 one-dimensional manifold, and each component can be described by a curve x ( ). In this note a theorem is proved which is directly related to and motivated by a result due to Eaves and Scarf on piecewise linear functions. This theorem relates the signs of the derivatives x t ( ) to the signs of the determinants of submatrices of the Jacobian matrix H '. Applications to solving nonlinear equations are given.