Multi-Variable Calculus Assignment

due Tuesday, April 24, 2001

Exercises

1. Let F be the vector field in the plane that is defined away from the origin by

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

Find the path integral of this vector field over the path C that is given parametrically by

R(t) = (r cos t, r sin t) for α ≤ t ≤ β

when r > 0 is a given constant.

2. Let C be the path that is given parametrically by

R(t) = ((cos(θ))t, (sin(θ))t) for 0 ≤ t ≤ r ,

where r and θ are constants. Evaluate the path integral

∫_{C} F

when F is the vector field that is defined by (a)

F (x, y) = (ye^{xy}, xe^{xy}) .

(b)

F (x, y) = (x − y, x + y) .

(c)

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

(d)

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