We first define the concepts of generalized characteristic coefficients and generalized Newton operators in mixed exterior algebra and study their properties. We then develop some computational techniques in mixed exterior algebra and prove a new inequality in mixed exterior algebra. Secondly, we apply mixed exterior algebra to the study of differential geometry. In particular, the concepts of divergences and Codazzi operator are developed. The fundamental result is an expression for the exterior derivatives of certain {dollar}(n - 1){dollar}-forms, involving Codazzi operator, on an n-dimensional manifold. If the manifold is compact and orientable, then Stokes' Theorem yields a very general integral formula. We show that, under specific geometric situations, this formula recovers many known integral formulae in differential geometry. As applications for the integral formula, we first characterize certain umbilic hypersurfaces with boundaries and secondly we find conditions under which a Riemannian manifold is locally flat.
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