Math

MATH 310, FALL 2017: ASSIGNMENT 6

Due: Wednesday, December 13, in class

Please answer the following questions, explaining all of the steps that you took to find your solutions. These problems cover material from Sections 13.4 and 13.5 in the textbook.

1 Evaluate ∫ C xy dx + x2 dy

along the boundary of the rectangle with vertices (0, 0), (3, 0), (3, 1) and (0, 1), oriented counterclockwise, in two different ways.

a) First do this directly. b) Second, use Green’s theorem to evaluate this as a 2-dimensional

the integral over a the rectangle {(x, y) : 0 ≤ x ≤ 3, 0 ≤ y ≤ 1}. 2 Use the method shown in class (Example 5, Section 13.4 of the text-

book) to determine the integral of∫ C F ·d~r

where

F = 2xy

(x2 + y2)2 i +

y2 −x2

(x2 + y2)2 j

and C is any positively oriented, simple curve which contains (0, 0). 3. Let f(x, y, z), g(x, y, z) be functions whose first order partial deriva-

tives exist, and F (x, y, z) = 〈P, Q, R〉 a vector field with P, Q, and R functions whose first order partial derivatives exist. Check that the following identities hold.

a) div(fF ) = f divF + F ·∇f. Here fF = 〈fP, fQ, fR〉. b) div(∇f ×∇g) = 0. c) curl(curl F ) = grad(div F ) −∇2F .