(a) Let C be a simple closed piecewise-smooth space curve which lies entirely in a plane, and suppose that the plane has upward-pointing unit normal vector given by n = ai + bj + ck. Show that the area of the portion of the plane enclosed by C is 1 2 C
(bz − cy) dx + (cx − az) dy + (ay − bx) dz.
(b) Let C be a simple closed smooth curve in the plane 2x + 2y + z = 2, oriented counter-clockwise as viewed from above. Show the the integral 2ydx + 3zdy − xdz C
depends only on the area of the region enclosed by C and not on the position or shape of C.
(0 points) A torus of revolution is the surface obtained by rotating a circle C in the xz-plane about the z-axis in space. Assume that the radius of C is r > 0 and the center of C is (R, 0, 0), where R > r. (a) Find a parameterization of the torus.
Find the surface area of the torus.
(0 points) Let f (x, y, z) and g(x, y, z) be scalar functions with continuous second partial derivatives, and let S be a surface with boundary curve C (where C is positively-oriented relative to S). Show that ( f× S
g) · dS =
f g · dr.
(0 points) Let F = P i + Qj + Rk be a vector ﬁeld whose components are continuously diﬀerentiable. We deﬁne the notation F · to mean P Prove that (F · G) = (G · ) F + (F · )G + F × ( × G) + G × ( × F) . ∂ ∂ ∂ +Q +R . ∂x ∂y ∂z
(0 points) Let F(x, y, z) be a vector ﬁeld on R3 − (0, 0, 0) given by F(x, y, z) = 2yz (x2 + y 2 + z 2 )2 i+ −2xz (x2 + y 2 + z 2 )2 j.
Let S be the closed surface which is the boundary of the region enclosed by the hyperboloid x2 + y 2 − z 2 = 4 and the planes z = 5 and z = −5 with outward orientation. Compute the ﬂux integral F · dS. S