13.2 Oriented Curves 1.

(a) A spiral on the elliptic cylinder y2 + 9z2 = 9 oriented clockwise when viewed from far out the x axis.

(b) A cubical parabola (it looks like a stylized gull in flight) on the plane z = x oriented from left to right when viewed from far out the plane y = x.

(c) A sine wave on the parabolic cylinder y = x2 oriented from right to left when viewed from far out the y axis.

(d) An ellipse sliced by the plane x = z out of the cylinder y2 + Z2 = 1 oriented clockwise when viewed from far out the x axis.

(e) A sine wave traced vertically on the plane y = x oriented from below to above when viewed from far out the x axis.

2.

(a) 128/3.

(b) -7rV2/2.

(c) 0.

3.

(a) 5.

(b) 7r(-1+V5)/2.

(c) IRI(2-a-b)/2.

(d) -sin(l) +1/3.

4.

(c) There exist functions 'Ij; and T on [0,1] that are CIon (0,1) \ {j /N : j =

1, ... ,N} such that T' > ° and 'Ij; = ¢j OT on ((j -l)/N,j/N) for each j =

1, ... ,N.

7.

(c) If F is conservative, consider the case when C is smooth first. If (*) holds, use parts

(a) and

(b) to prove that F is conservative.

8. Use Jensen's Inequality.