Mathematics 173 – Applied Mathematics 1 Problem Set 2 – Line and Surface Integrals September 14, 1999

I. Evaluate the line integrals of each of the following vector fields along the indicated curves, and answer the accompanying questions.

A) F = (x, x + y), α(t) = (t − t^{2}, t^{2} + t^{3}), t ∈ [0, 1]. Answer: 139/60. B) F = (y^{2}, 2xy), α the path from (−1, 2) to (1, 5) made up of two straight line segments,

the first parallel to the x-axis, the second parallel to the y-axis. C) F = same vector field as in part B, but now α the line segment directly from (−1, 2)

to (1, 5). D) Your answers for B and C should be equal. Explain why this is true. Hint: Is F a

gradient vector field? E) The vector field

F = (\frac{−y}{x^{2} + y^{2}} , \frac{x}{x^{2} + y^{2}} )

α the circle of radius 1 centered at (0, 0) in R^{2}, traversed counterclockwise.

II. Use Green’s Theorem to evaluate each of the following line integrals around the specified simple closed curve, and answer the other questions.

A) F = (sin(πxy), cos(πxy)), α the positively-oriented boundary of the square [0, 1] ×

[0, 1]. Answer: −\frac{2+π}{π}. B) F = (y^{2}, xy), α the ellipse with equation x^{2}/9 + y^{2}/4 = 1. C) Compute

∫∫_{D} \frac{∂Q}{∂x} − \frac{∂P}{∂y} dA

for the vector field F and the region D inside α from part E of question I. Your answer should be different from what you saw in that part. Is this a contradiction to Green’s Theorem?? Explain.

III. The statement of Green’s Theorem is also valid in more general situations. For example, consider the circular ring D in the plane between the two circles x^{2}+y^{2} = 1 and x^{2}+y^{2} = 4. The positively oriented boundary of D consists of the the circle of radius 2 traversed counterclockwise, and the circle of radius 1 traversed clockwise. Let F = (P, Q) be a vector field whose component functions have continous first order partial derivatives on some open set in R^{2} containing the whole ring. Show that the line integral of F over the positively oriented boundary is equal to the double integral of \frac{∂Q}{∂x} − \frac{∂P}{∂y} over the ring. Hint: Consider the curve α_{1} which consists of the line segment from (1, 0) to (2, 0), followed by the upper semicircle of radius 2 traversed counterclockwise, followed the line segment from (−2, 0) to (−1, 0), followed by the upper semicircle of radius 1 traversed clockwise. Similarly let α_{2} be the curve consisting of the line segment from (−1, 0) to (−2, 0), followed by the lower semicircle of radius 2 traversed counterclockwise, followed by the line segment from (2, 0) to (1, 0), followed by the lower semicircle of radius 1 traversed clockwise.

IV. For each of the following vector fields and surfaces, verify the Divergence Theorem by direct computation of the surface integral and the triple integral: A) F = (z, y, x), S the sphere of radius 1 centered at (0, 0, 0). B) F = (x + y, y + z, x + z), S the surface of the tetrahedron bounded by the coordi-

nate planes and the plane x + y + z = 1. (For the surface integral, you will need to parametrize each of the four faces of the tetrahedron separately, and add the corresponding surface integrals.) Answer: 1/2 (both values).

Assignment

Individual solutions, due Tuesday, September 21.