MONICA K. HURDAL Teaching Phone: +1 850 644-7183    Fax: +1 850 644-4053 E-mail: mhurdal@math.fsu.edu

Solutions:
Q1: Circulation = -1/6.
Q2a): Vectors point upward for y > 0, point downward for y < 0; increase in length (magnitude) as |y| gets larger.
Q2b): Positive, since the vector field is flowing in the same direction as C₁.
Q2c): Zero, since the fector field is flowing perpendicular to C₂.
Q2d): Zero, due to symmetry in the vector field and C₃.
Q2e): Yes, since the line integral between 2 points seems to be the same regradless of the path, and the line integral of a closed curve (C₃) ise zero, indicitating path independence. Also, the potential function can be computed, which is f(x,y) = y²/2 + C.
Q3: Parameterize the curve with x = 2cos(t)+2, y= 2sin(t)+2, z=2 for t from 0 to Pi/2. Work = -4.
Q4: Flux = 24. Use the formula from 19.2 where the surface is of the form z = f(x,y).
Q5a): curl F = 0 and F has no holes, so F is path-independent, which means F is a gradient vector field.
Q5b): Line integral = 25/2 where f(x,y) = xy² - cos(xy) - y²/2 + C
Q6a): x² + y² = s² = z² and since s is positive, then z = sqrt(x² + y²) for z between 1 and 2. Thus, this surface is a frustrum (ie. a cone with vertex at the origin, but only the part of the cone for z between 1 and 2).
Q6b): Flux = -15Pi/4. You can either use the formula from 19.3 or the formula from 19.2 where the surface is of the form z = f(x,y).
Bonus: Compute curl G and you will find curl G = curl F. Since F is path-independent, curl F = 0. Thus curl G = 0, and so, G is also path-independent.