208 Math Homework 6
due on Thursday 4/30/09 Problem 1. Verify the product rule
( f G) = f (
G) +
f G,
where f is a smooth scalar function and G is a smooth vector eld in R3 . Conclude that ( f g) = f g for any pair f , g of smooth scalar functions. Problem 2. Use Stokes Theorem to evaluate (i)
(
F) dS, where F( x, y, z) = y i + z j + x k and the surface is the
portion of the paraboloid z = 1 x2 y2 with z 0, oriented with upwardpointing normal. (ii)
C y2 + z2
y dx + z dy + x dz, where the circle C is the intersection of the sphere x2 +
= R2 and the plane x + y + z = 1, oriented counterclockwise viewed as from above.
Problem 3. Let F( x, y, z) = a ( x i + y j + z k), where a R3 is a xed vector. Suppose is a surface with boundary C which satises the conditions of Stokes Theorem. Show that F ds = 2 a d S.
C
[This situation arises, for example, when F and a are the velocity and angular velocity of a rotating rigid body, respectively.] Problem 4. Let Ch de...