WORKSHEET XIV
1. Verify Green's Theorem for each of the following vector fields and regions:
(A) F(x,y) = (x-y) i + x j and R is the unit disk centered at the origin.
(B) G(x,y) = y2i
+ x2 j and T is the triangle with vertices
(0,0), (1,2) and (2,5).
2. Using Green's Theorem, evaluate:
where S is the unit square bounded by x=0, x=1,
y=0, y=1.
3. State the area formula that is a consequence of Green's Theorem. Using this formula, find the area of the ellipse r(t) = (a cos t) i + (b sin t) j, 0<t<2p.
4. Using Green's Theorem, find the work done by the vector field F = 2xy3 i + 4x2y2j in moving a particle once counterclockwise around the triangular region in the first quadrant bounded by the x-axis, the line x=1, and the curve y = x3.
5. State the general version of Green's Theorem for regions that may contain holes. Let R be the annulus in the plane given by 1 < x2 + y2 < 4. Let F(x,y) be the vector field (x2 + y2)-1(-y i + x j). (This may be used to model the velocity field of a tornado.) Verify Green's Theorem for F on the annulus.
6. State the Divergence Theorem in the plane. Verify the theorem for the vector field F(x,y) = x i + y j across the unit disk, D, centered at the origin.
7. Verify the Divergence Theorem in the plane for the field G(x,y) = y3 i + x5j across the square bounded by the lines x=0, x=3, y=1, y=4.
8. Using the divergence theorem in the plane, calculate the net flux of the given vector field over the given boundary:
(A) F(x,y) = x i + y2j across the square bounded by the lines x=+1, y=+1.
(B) G(x,y) = (y2 - x2) i + (x2 + y2) j across the triangle bounded by the lines y=0, x=1, and y=x.
(C) H(x,y) = xy i + y2j over the region enclosed by the curves y = x2 and y = x in the first quadrant.
(D) A(x,y) = (x-y) i + x j across the circle x2 + y2 = 1.
(E) B(x,y) = 2xy i
- y2 j through the ellipse (x/a)2
+ (y/b)2 = 1.
GREEN'S
MILL: Once home of the mathematical physicist, George
Green (1793-1841)