30.6. EXERCISES 609

11. Find∫

∂U F ·dR where U is the set {(x,y) : 1 ≤ x ≤ 2,x ≤ y ≤ 3} and

F (x,y) = (xsiny,ysinx)

12. Find∫

∂U F ·dR where U is{(x,y) : x2 + y2 ≤ 2

}and F (x,y) =

(−y3,x3

).

13. Show that for many open sets in R2, Area of U =∫

∂U xdy, and Area of U =∫

∂U −ydxand Area of U = 1

2∫

∂U −ydx+ xdy. Hint: Use Green’s theorem.

14. Two smooth oriented surfaces, S1 and S2 intersect in a smooth oriented closed curveC. Let F be a C1 vector field defined on R3. Explain why

∫S1

curl(F ) ·ndS =∫S2

curl(F ) ·ndS. Here n is the normal to the surface which corresponds to thegiven orientation of the curve C.

15. Show that curl(ψ∇φ) = ∇ψ ×∇φ and explain why∫

S ∇ψ ×∇φ ·ndS =∫

∂S (ψ∇φ) ·dr.

16. Find a simple formula for div(∇(uα)) where α ∈ R.

17. Parametric equations for one arch of a cycloid are given by x = a(t − sin t) and y =a(1− cos t) where here t ∈ [0,2π]. Sketch a rough graph of this arch of a cycloidand then find the area between this arch and the x axis. Hint: This is very easy usingGreen’s theorem and the vector field F = (−y,x).

18. Let r (t) =(cos3 (t) ,sin3 (t)

)where t ∈ [0,2π]. Sketch this curve and find the area

enclosed by it using Green’s theorem.

19. Verify that Green’s theorem can be considered a special case of Stoke’s theorem.

20. Consider the vector field(

−y(x2+y2)

, x(x2+y2)

,0)= F . Show that ∇×F = 0 but that

for the closed curve, whose parametrization is R(t) = (cos t,sin t,0) for t ∈ [0,2π],∫C F ·dR ̸= 0. Therefore, the vector field is not conservative. Does this contradict

Theorem 30.5.7? Explain.

21. Let x be a point of R3 and let n be a unit vector. Let Dr be the circular disk of radiusr containing x which is perpendicular to n. Placing the tail of n at x and viewingDr from the point of n, orient ∂Dr in the counter clockwise direction. Now supposeF is a vector field defined near x. Show that curl(F ) ·n = limr→0

1πr2

∫∂Dr

F ·dR.This last integral is sometimes called the circulation density of F . Explain how thisshows that curl(F ) ·n measures the tendency for the vector field to “curl” aroundthe point, the vector n at the point x.

22. The cylinder x2 + y2 = 4 is intersected with the plane x+ y+ z = 2. This yields aclosed curve C. Orient this curve in the counter clockwise direction when viewedfrom a point high on the z axis. Let F =

(x2y,z+ y,x2

). Find

∫C F ·dR.

23. The cylinder x2 + 4y2 = 4 is intersected with the plane x+ 3y+ 2z = 1. This yieldsa closed curve C. Orient this curve in the counter clockwise direction when viewedfrom a point high on the z axis. Let F =

(y,z+ y,x2

). Find

∫C F ·dR.