(906, #27) Let F be the following vector field:

a) If C₁ is the vertical line segment from ( -3, -3 ) to ( -3, 3 ), determine whether

is positive, negative, or zero.

b) If C₂ is the counterclockwise-oriented circle with radius 3 and center the origin, determine whether

is positive, negative, or zero.

Solution:

a) Along the line x = -3, the path we follow is upward. The angle created by the vectors of F and the upward path we are following are smaller than ninety degrees. The cosine value of a number smaller than ninety degrees is positive. Therefore, the integrand is positive.

b) The counterclockwise-oriented path along a circle of radius 3 is going in the opposite direction of the vectors of F. The angle created by F and the tangents to the circle are obtuse, therefore the integrand is negative.

By C.A.H.