Question

Determine whether the following are true or false: A) If Sis a surface parametrized byr:DR^3, then A(S) = (double integral)D dA, where A(S) is the surface area of S. B) Let c be a boundary of a closed and bounded region D in the xy-plane. Then counterclockwise is always a positive orientation of c. c) Let Fbe a constant vector field on R^3. Then the flux of F through the unit sphere x^2 + y^2 + 2^2 = 1 is always zero. D) 0 (t) = 6 - et is an orientation-reversing reparametrization. E) It is possible to use Green's Theorem to calculate the path integral (integral)c F.ds, where F(x, y) = (-y/(x 2 + y2), x/(x 2 + y 2 )) and c is the unit circle x 2 + y 2 - 1, oriented positively? IF

Answer 1

# SOLUTION! (A) IF S is a surface parametrized by then ACS) = V:D R², Supturtu da & Soda where Also is the surface area of s. So, A IS FALSE. (B) let C be a boundary of a luwid and boundid Suglon in ay Plan thin counterflochure is always a Paritive arientation of C. TRUE. we know that curve (has a partrive orientation if it is traced out in a counter - Jock wire direction. (C) Let F be a constant vcrus fild our3 Hun the filux of F through ou unit spruce x² + y2 + 2? = 1 ks always zuro. - TRUE. Because, Flux = SS & de 0) (1) d (t) = 6-et is an orientation - ruwering rue parametrization we knew that A suparamitorization (1) of a tuwe is ovuiqutuliou Beserving if Ø (e) >0 and aun notion is durowing if &' (t) so. Here di(t) = -et co (E) It is Parrible to use verun's theorum to calewate the Path integral de Fides, where Flky) = (127 x2+42 72 +42 472) and is the unit will ?? + y2 = 1, ariented Pasitiuly. - TRUE. > 2 & TRUE. since. To Seridosso ( 29 )