Let g: A →Rp be as in Theorem 5-1. If f: Rn → R is differentiable and the maximum (or minimum) of f on g-1 (0) occurs at , show that there are , such that
Answer to relevant Questionsa. Let Ί: Rn → Rn be self-adjoint with matrix A = (aij), so that aij = aji. If f (x) = =Σ aij xixj, show that Dkf (x) = 2 Σj = 1 akjxj. By considering the maximum of on Sn-1 show that there is ...If M1CRN is an -dimensional manifold-with-boundary and M 2 C M1 - ∂M1 is an -dimensional manifold with boundary, and M1, M2 are compact, prove thatIf there is a nowhere-zero k-form on a k -dimensional manifold M, show that M is orientableFor the triangular element in Fig P1.3, show that a tilted free liquid surface, in contact with an atmosphere at pressure pa, must undergo shear stress and hence begin to flow. For low-speed (laminar) flow in a tube of radius ro, the velocity u takes the form u = B Δp/μ (r2- r2) Where μ is viscosity and Δp the pressure drop. What are the dimensions of B?
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