Question

Suppose that the vector field, $F=F(x,y,z)\epsilon \mathbb{R}^3$ , is continuously differentiable and satisfies in the interior of the domain , open and bounded, whose boundary $\partial \Omega$ is a smooth surface (at least $C^1$ class) , steerable. Show that cannot be tangent to $\partial \Omega$ in every point of the surface $\partial \Omega$

Answer 1

Given A z that for the vector field F = F(x,y; 7) is continuously differentiable and dirt du fi tay a F2 + d2 F3 2.0 t (1,4,Z)en whose boundary is smooth If possible let, f is tangent to on in every point of an If û is the normal (unit and outward) to as then ou 20 H. (91,9,2) 0,2 FI How by by dive SI divergewe theorem, F. ñ ds. SSS div Ēdr &n vise NO 2 SIS divĚ dv بار . contradiction as div ř 7o and volume of 270, WW: divť dvo est nowher F can not be Hence, the surface on tangent to as in every point of