Let S be an orientable surface with unit normal n and nonempty boundary Iuml S which satisfies the hypotheses of Stokes 's Theorem a) Suppose that F S R3 0 is Cl, that Iuml S is smooth, and that T is the unit tangent vector on Iuml 5 induced by n If the angle between T(x0) and F(x0) is never obtuse...

a Let be the angle between F and T Since 0 2 cos 0 ie F T 0 Hence by Stokess The ...

Let S be an orientable surface with unit normal n and nonempty boundary ÏS which satisfies the

Question:

Let S be an orientable surface with unit normal n and nonempty boundary Ï‘S which satisfies the hypotheses of Stokes's Theorem.
a) Suppose that F: S †’ R3{0} is Cl, that Ï‘S is smooth, and that T is the unit tangent vector on Ï‘5 induced by n. If the angle between T(x0) and F(x0) is never obtuse for any x0 ˆˆ Ï‘5, and ˆ«ˆ«S curl F ˆ™ ndÏƒ = 0, prove that T(x0) and F(x0) are orthogonal for all x0 ˆˆ Ï‘S.
b) If Fk,: S R3 are C1 and Fk †’ F uniformly on Ï‘S, prove that