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10 Nov 2019
Differential Calculus
(9 marks) Let x: [0,1] rightarrow Rn be infinitely many times differentiable. Assume that x'(s) 0 [0,1]. Define (1 mark) Let g(s) = ||x'(S)|| [0,1]. Prove that (1 mark) Prove that T'(s).T(s) = 0 (2 marks) Let K(s) be the curvature of {x(u) ; u [0,1]} at point x(s). Prove that ||T'(S)|| = ||x'(S)||K(s) [0,1]. (1 mark) Prove that T''(s).T(s) = -||T'(s)||2 [0,1]. (1 mark) Assume that T'(S) 0 [0,1]. Define Prove that (3 marks) Consider the equation where T(t,s) = N(t,s) = K(t,s) = for all t 0 and s [0,1], and x: [0, infinity] times [0,1] rightarrow Rn is an infinitely differentiable solution such that and are non zero for all t 0 and all s [0,1]. Define Assume that . Prove that the length of {x(t,s) ; s [0,1]} decreases as t increases. Hint: .
Differential Calculus
(9 marks) Let x: [0,1] rightarrow Rn be infinitely many times differentiable. Assume that x'(s) 0 [0,1]. Define (1 mark) Let g(s) = ||x'(S)|| [0,1]. Prove that (1 mark) Prove that T'(s).T(s) = 0 (2 marks) Let K(s) be the curvature of {x(u) ; u [0,1]} at point x(s). Prove that ||T'(S)|| = ||x'(S)||K(s) [0,1]. (1 mark) Prove that T''(s).T(s) = -||T'(s)||2 [0,1]. (1 mark) Assume that T'(S) 0 [0,1]. Define Prove that (3 marks) Consider the equation where T(t,s) = N(t,s) = K(t,s) = for all t 0 and s [0,1], and x: [0, infinity] times [0,1] rightarrow Rn is an infinitely differentiable solution such that and are non zero for all t 0 and all s [0,1]. Define Assume that . Prove that the length of {x(t,s) ; s [0,1]} decreases as t increases. Hint: .