A Theorem on Homotopy Paths

We consider the set of points x ∈ Rⁿ⁺¹ satisfying H(x) = 0, where H: Rⁿ⁺¹ → Rⁿ is a C¹ function and 0 is a regular value. This set, H⁻¹(0), is a C¹ one-dimensional manifold, and each component can be described by a curve x(θ). 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(θ) to the signs of the determinants of submatrices of the Jacobian matrix H′. Applications to solving nonlinear equations are given.