Question: Let f: R2 - R be a smooth function. Assume that f(0) = 0 and (Vf) (0) = 0. Let fxx(0) fxy(0) Hi = fry

Let f: R2 - R be a smooth function. Assume that f(0) = 0 and (Vf) (0) = 0. Let fxx(0) fxy(0) Hi = fry (0) fyy (0) be its Hessian at 0. Let If C R3 be the graph of f. Consider the regular parameterization r(u, v) = (u,v, f(u, v) ) of If. Let B = {ru(0), rv(0) } be the induced basis of of ToIf. 2.1. 5 points. Show that Hy is the matrix of the shape operator of If at 0 with respect to B. 2.2. 2 points. Conclude from the previous part that the Gaussian curvature of If at 0 is det(Hf)

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