Let d da b fxx a bfyy a b fxy a b2 a if

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Unformatted text preview: ntinuous on a disk with center (a, b) and suppose ∇f (a, b) = (0, 0). Let D = D(a, b) = fxx (a, b)fyy (a, b) − fxy (a, b)2 . (a) If D > 0 and fxx (a, b) > 0, then (a, b) is a local minimum. (b) If D > 0 and fxx (a, b) < 0, then (a, b) is a local maximum. (c) If D < 0, then (a, b) is a saddle point. Curves: r(t) = (x(t), y (t), z (t)) t ||r ′ (u)|| du Arclength s(t) := (measuring from a to t) a Unit Tangent Vector T (t) := Curvature Vector r ′ (t) ||r ′ (t)|| T ′ (t) dT = ds ||r ′ (t)|| Curvature κ := dT ds Principle Unit Normal N (t) := 1 dT = κ ds T ′ (t) T ′ (t) Binormal B (t) := T (t) × N (t) B (t) = r ′ × r ′′ ||r ′ × r ′′ || 2 d2 s a( t ) = 2 T + dt aT = ds dt r ′ • r ′′ ||r ′ || ||r ′ × r ′′ || ||r ′ || aN = κ N = aT T + aN N 2 κ= ||r ′ × r ′′ || ||r ′ ||3...
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This document was uploaded on 02/15/2014 for the course MATH 222 at Kansas State University.

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