7. (7 points) True-false questions. Circle T if the given statement is true, and circle F if it is false.

There is no need to justify your answer.

(a) T F The surface given by x² + y² = x in R³ is the same as r = cos in cylindrical

coordinates.

(b) T F If F(x, y, z) = (z, x, -y), then curl(F) is perpendicular to the vector (1, 0, 1).

(c) T F The object given by r(u) = (cos u, sin u, 1) represents a cylinder in R³.

(d) T F p² sin dedødp tells us the volume of a half-ball of radius 1.

π/2 2π

[²[*/² [²* o

2

0

(e) T F If C₁ is an oriented curve in R2, and C₂ is the same curve with the opposite orien-

- SC₂ x² ds

C2

tation, then

Jo

C₁

2

x² ds

(f) T F For any smooth function f(x, y), if the curves C₁ and C₂ have the same initial point

Vf. dr

and terminal point, then

Vf. dr =

C₁

C2/n(e) T F If C₁ is an oriented curve in R2, and C2 is the same curve with the opposite orien-

- Jo₂²

tation, then

C₁

x² ds = -

2

x² ds

(f) T F For any smooth function f(x, y), if the curves C₁ and C₂ have the same initial point

and terminal point, then

Vf. dr = Vf. dr

C₁

C2

(g) T F The surface given by x² + y² = x in R³ is the same as r = cos in cylindrical

coordinates.

(h) T F Let F(x, y, z) = (0, 0, z) and S be the sphere x² + y² + z² = 1 with outward pointing

normals. The flux integral F Fn dS must be positive.