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Let U be an open subset of R2, 0 EU, f: UR be a differentiable function, and Vƒ(0,0) = (0,0). Let S be the regular surface given by the graph of f. A parametrization of S is: X(u, v) = (u, v, f(u, v)), (u, v) ≤ U. Show that at p = X(0,0), the second fundamental form of S, with respect to the upward pointing unit normal N, is given by the Hessian matrix at (0,0). [fuu fuv] [fuv fvv]

Let U be an open subset of R2, 0 EU, f: UR be a differentiable function, and Vƒ(0,0) = (0,0). Let S be the regular surface given by the graph of f. A parametrization of S is: X(u, v) = (u, v, f(u, v)), (u, v) ≤ U. Show that at p = X(0,0), the second fundamental form of S, with respect to the upward pointing unit normal N, is given by the Hessian matrix at (0,0). [fuu fuv] [fuv fvv]

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let U be an open subset of R², 0 € U, ƒ : U → R be a differentiable function, and Vƒ(0,0) = (0, 0). Let S be the regular surface given by the graph of f. A parametrization of S is: X(u, v) = (u, v, f(u, v)), (u, v) ≤ U. = Show that at p X(0, 0), the second fundamental form of S, with respect to the upward pointing unit normal N, is given by the Hessian matrix at (0,0). [fuu fuv [ fuv fvv_
Transcribed Image Text:Let U be an open subset of R², 0 € U, ƒ : U → R be a differentiable function, and Vƒ(0,0) = (0, 0). Let S be the regular surface given by the graph of f. A parametrization of S is: X(u, v) = (u, v, f(u, v)), (u, v) ≤ U. = Show that at p X(0, 0), the second fundamental form of S, with respect to the upward pointing unit normal N, is given by the Hessian matrix at (0,0). [fuu fuv [ fuv fvv_
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