Answer to Question #199502 in Geometry for Mahnoor

Question

Let p,q∈R

n

p,q∈Rn and let γ

γ be a curve such that γ(a)=p,γ(b)=q

γ(a)=p,γ(b)=q, where a

a < b

b.

(a) Show that, if u

u is a unit vector, then

γ

˙

⋅u≤∥γ

˙

∥

γ˙⋅u≤‖γ˙‖

(b) Show that

(q−p)⋅u≤∫

a

b

∥γ

˙

∥dt

(q−p)⋅u≤∫ba‖γ˙‖dt

(c) Show that the arc length ofγ

γfromγ(a)

γ(a)toγ(b)

γ(b)is at least∥q−p∥

‖q−p‖, with equality whenγ

γis a straight line.

Answer 1

Let f be a continuous complex-valued function of a complex variable, and let C be a smooth curve in the complex plane parametrized by. Z(t) = x(t) + i y(t) for t varying between a and b. Then the complex line integral of f over C