Regular Equilibria. A First Approach
In this section we consider equilibria of gradient fields only. An equilibrium of v = grad f is a point x where Txf = o, i.e. a critical point of f. A mapping may have many critical points, but according to Sard’s theorem it has only few critical values if it is smooth enough. We shall need and prove Sard’s theorem only in a simple special case. An application then yields that excess demand, if it is a gradient field, has “in general” only regular and therefore isolated equilibria.
KeywordsGradient Field Excess Demand Countable Union Differentiable Manifold Countable Base
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