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Problem 2.7.5 (Gaussian Curvature). Let f:R R be a smooth function. Consider a surface S = {(x1, ... ,Xn, f(x1, ... ,xn)) : (x1, ... ,Xd) E U} for an open domain U CR. For a R". Set the hessian H(a) = (fx;x;(a)). Then we call det H(a) the Gaussian curvature of S at a. Since H(a) is a symmetric matrix, it follows that H(a) is orthogonally diagonalizable and H(a) with its eigenvalues 2; and eigenvectors H(a) = 2;v; for I = 1, ... ,n. Let (aj, a2) RP. (a) Prove that 2; is the curvature of the surface at a along vi by showing that () *r(80)_0 = ; where yo) = a and Y(0) = vi for example y(t) = a + vit. (b) Find the Gaussian curvatures and the curvature along vi for each i = 1,2 at a (Consider eigenvalue li of the eigenvector yi) and illustrate the curvatures by draw- ing the figures near (a, f(a)) of the three surfaces below f(x1,x2) = x} + x f(x1, x2) =x7-13 f(x1,x2) = |(x1,x2) . Problem 2.7.5 (Gaussian Curvature). Let f:R R be a smooth function. Consider a surface S = {(x1, ... ,Xn, f(x1, ... ,xn)) : (x1, ... ,Xd) E U} for an open domain U CR. For a R". Set the hessian H(a) = (fx;x;(a)). Then we call det H(a) the Gaussian curvature of S at a. Since H(a) is a symmetric matrix, it follows that H(a) is orthogonally diagonalizable and H(a) with its eigenvalues 2; and eigenvectors H(a) = 2;v; for I = 1, ... ,n. Let (aj, a2) RP. (a) Prove that 2; is the curvature of the surface at a along vi by showing that () *r(80)_0 = ; where yo) = a and Y(0) = vi for example y(t) = a + vit. (b) Find the Gaussian curvatures and the curvature along vi for each i = 1,2 at a (Consider eigenvalue li of the eigenvector yi) and illustrate the curvatures by draw- ing the figures near (a, f(a)) of the three surfaces below f(x1,x2) = x} + x f(x1, x2) =x7-13 f(x1,x2) = |(x1,x2)

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