Multi-Variable Calculus Assignment

due Friday, May 4, 2001

Exercises

1. Let F be the vector field given by

F (x, y, z) = [e^{x}(cos y)z^{2}, −e^{x}(sin y)z^{2}, 2e^{x}(cos y)z] .

Find the path integral of F over the polygonal path consisting of the line segment from the origin to the point (2, −1, 3) followed by the line segment from that point to the point (0, 0, 1).

2. Find the surface integral of the vector field

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

over the boundary surface of the domain

4 < x^{2} + y^{2} + z^{2} < 9

when that surface is oriented by its exterior normal.

3. Find the curl of the vector field

F (x, y, z) = (z^{2}x, x^{2}y, y^{2}z) .

4. If S is any piece of surface in space whose oriented boundary is the boundary of the first quadrant portion of the disk

x^{2} + y^{2} ≤ a^{2}, z = 0 ,

traversed counter-clockwise with respect to the standard orientation of the horizontal plane z = 0, find the integral over S of the vector field

(yz, zx, xy) .