Solutions #6
1(a). For a vector eld F : R3 R3 , show that
a vector eld is incompressible.
(
F ) = 0; in other words, the curl of
Solution. If F (x, y, z) = F1 (x, y, z) + F2 (x, y, z) + F3 (x, y, z)k, then we have
x
(
y
z
F ) = det x y z
F1 F2 F3
F3
Solutions #2
1a. Consider the surface in R3 determined by the equation
z sin(x + y) + x2 3xy + 2z = 7.
Find a function g(x, y, z) such that this surface is a level set of g and nd a function f (x, y)
such that this surface is the graph of f .
Solution. g(
Solutions #5
1. Consider the surface dened by the equation
x3 z + x2 y 2 + sin(yz) = 3 .
(a) Find an equation for the plane tangent to this surface at the point (1, 0, 3).
(b) Parametrize the line normal to this surface at the point (1, 0, 3).
Solution.
(
Solutions #4
1.(a). Let F , G : Rn R3 be dierentiable at a Rn and let x Rn . Show that
D(F G)(a) x = DF (a) x G(a) + F (a) DG(a) x .
Solution. Since both sides of the identity depend linearly on x, it is sucient to prove it for
x = e1 , x = e2 , . . . , x
Solutions #8
1. (a) Find the volume of an ice cream cone bounded by the cone z = x2 + y 2 and the
hemisphere z = 8 x2 y 2 .
(b) Find the average distance to the origin for points in the ice cream cone region bounded
by the hemisphere z = 8 x2 y 2 and the
Solutions #9
1(a). Suppose that : [a, b] R3 with (t) := x(t) + y(t) + z(t)k is a smooth parameterization of the curve C with endpoints p := x(a), y(a), z(a) and q := x(b), y(b), z(b) . Let
f : R3 R be a smooth function. If h : R R is the composite functio
Solutions #11
2
1. Evaluate Q E dS where E(x, y, z) := zex + 3y + (2 yz 7 )k and Q is the union of the
ve upper faces of the unit cube [0, 1] [0, 1] [0, 1] orient outward. The face z = 0 is not
part of Q.
Solution. We write W := [0, 1] [0, 1] [0, 1] for t
Solutions #10
1. (a) Find a parametrization for the hyperboloid x2 + y 2 z 2 = 25;
(b) Find an expression for a unit normal to this surface.
(c) Find an equation for the plane tangent to the hyperboloid at the point (a, b, 0) where
a2 + b2 = 25.
(d) Show
Solutions #7
1. Evaluate
x + y dA,
D
where D is the region in the rst quadrant of the plane bounded by the lines y = x and
y = 3x and the hyperbola xy = 3.
Solution. It is not dicult to see that D is given by the inequalities
0 x
3
x y (x),
where
(x) =
H
