Two adjacent regions share one edge. Their union has no hole. Their positively oriented boundary circulations for the same smooth field are 4 and -1. What is the positive circulation around the union's exterior boundary?

P and Q have continuous first partial derivatives near an annulus and both boundary circles. I_outer and I_inner are both circle integrals taken counterclockwise. Which expression equals the double integral of Q_x - P_y over the annulus?

• I_outer + I_inner, using both given directions
• I_inner - I_outer, reversing only the outer curve
• I_outer - I_inner, reversing only the inner curve
• I_outer alone, omitting the excluded hole's boundary

Under Green's hypotheses, what equals ∮C(P dx + Q dy) when C is the positively oriented complete boundary of D?

• Double integral of Q_x - P_y over D
• Double integral of P_x + Q_y over D
• Double integral of Q_y - P_x over D

Green's theorem can be applied directly to any open arc without adding another curve to close the boundary.

Under Green's hypotheses, which choice makes ∮C(P dx + Q dy) equal the area of D when C is its positively oriented complete boundary?

• P = x/2 and Q = y/2, giving Q_x - P_y = 0
• P = y/2 and Q = -x/2, giving Q_x - P_y = -1
• P = -y/2 and Q = x/2, giving Q_x - P_y = 1

Around a simple closed curve traversed once clockwise, ∮(x dy - y dx) = -14. What is the enclosed area?

For F = (-y, 2x), the circulation around a positively oriented simple closed boundary is 27. Green's hypotheses hold. What is the enclosed area?

A field's circulation once counterclockwise around a simple closed curve is 5. What is its circulation once clockwise around the same curve?

• 5: orientation does not change a line integral
• 0: opposite orientations are added together
• -5: reversing the path negates its integral

A field is smooth near a simple closed curve but blows up at an enclosed point. A student applies Green's theorem to the entire filled region. Which check addresses the problem?

• Use the theorem because all field values on the boundary are finite
• Check the interior condition; the filled region contains a singularity
• Check the boundary alone and omit the singular point without adding any boundary

For F = (-y, x), C is the positively oriented complete boundary of a region D satisfying Green's hypotheses. What is Q_x - P_y, and how does circulation relate to area?

• It is 2, so circulation is twice the area
• It is 0, so circulation vanishes for every such region
• It is -2, so circulation is minus twice the region's area