1
Lecture #1: Vectors and the Scalar Product
1.1
Vector Algebra
vector in Rn : an n-tuple of real numbers
v = a1 , a2 , . . . , an .
For example, if n = 2 and a1 = 1 and a2 = 1, then w = 1, 1 is vector in R2 . Vectors are
represented by directed line seg

Selected Solutions to Homework # 5
#10 in 10.2: r (t) = (t3 + 4t)i + tj + 2t2 k, r(0) = i + j
Integrate once:
r(t) = (
t4
t
2t3
+ 2t2 )i + j +
k+C
4
2
3
Then solve for C by plugging in the initial value: r(0) = 0 + C if we
use the formula above. But we a

Selected Solutions to Homework # 6
#10 in 11.1: Find the domain and range of f . Describe the lelvel
curves. Find the boundary of the domain. Determine if the domain is
open, closed, both, or neither. Decide if f is bounded or unbounded.
f (x, y) = e(x
2

Selected Solutions to Homework # 4
1. Solve exercise # 16 in section 10.1.
Compute the angle between the velocity and the acceleration vectors
at time t = 0 where the position is given by
2
2
r(t) =
t i+
t 16t2 j.
2
2
Compute the velocity and acceleration

Selected Solutions to Homework # 10
# 32 in 11.5: The derivative of f (x, y, z) at a point P is greatest in the
direction of v = i + j k. In this direction, the value of the derivative
is 2 3.
(a) What is
f at P?
We are given that if u = v/|v|, then Du f

Selected Solutions to Homework # 9
# 8 in 11.4: Compute fu and fv for the following function:
f (x, y) = tan1 (x/y),
x = u cos v,
y = u sin v.
The chain rule states that
fu = fx xu + fy yu and fv = fx xv + fy yv .
We compute:
fx =
1
1
,
2 y
1 + (x/y)
fy

Selected Solutions to Homework # 7
# 36 in 11.2: Show that the following limit does not exist:
x4
.
(x,y)(0,0) x4 + y 2
lim
Consider the limit when x = 0 and y approaches 0. This limit exists
and is equal to 0.
On the other hand, cosider the limit when x

Selected Solutions to Homework # 11
#2 in 11.6: Find the equation of a tangent plane and the equation of
a normal line to the surface
x2 + y 2 z 2 = 18
at the point P (3, 5, 4).
Let f = x2 +y 2 z 2 . Then the surface is a level surface of f . Therefore,

Selected Solutions to Homework # 12
Find the local extrema and saddle points of
f = 3x2 + 6xy + 7y 2 2x + 4y.
We compute the partial derivatives to nd the critical points:
fx = 6x + 6y 2,
fy = 6x + 14y + 4.
Solving fx = 0 and fy = 0, we nd that
6x = 2 6y

Selected Solutions to Homework # 2
For each quadratic equation below,
(a) identify the curve as a parabola, an ellipse, or a hyperbola;
(b) identify the focus and directrix if it is a parabola, the two focal
points and the semi-major axis if it is an elli

Selected Solutions to Homework # 1
1. Let P = (1, 0), Q = ( 3 , 23 ), and R = ( 1 , 23 ). Prove that the quad2
2
rangle with vertices O, P , Q, and R is a rhombus. Then prove, using
vectors, that P R is orthogonal to OQ.
A rhombus is, by denition, a quadr

LB 220 Calculus III, Quiz 8 Solutions
x = r cos = sin cos ,
r = sin ,
y = r sin = sin sin
x2 + y 2 + z 2 = r2 + z 2 = 2
z = cos ,
dV = dx dy dz = r dz dr d = 2 sin d d d
1. Suppose that a cylindrical hole is removed from a sphere. More precisely,
suppose

LB 220 Calculus III, Quiz 9 Solutions
1. (3 points) Let C be the closed curve consisting three line segments: (0, 0)
to (2, 0) to (0, 1) and back to (0, 0). Parametrize each of these three line
segments. (You will be using these in the problems which foll

Solutions to Homework 5
1. Let z = f (x, y ) be a twice continuously dierentiable function of x
and y . Let x = r cos and y = r sin be the equations which
transform polar coordinates into rectangular coordinates. Show that
2z
2z
2z
1 2 z 1 z
+ 2= 2+ 2 2+

LB 220 Homework 4 Solutions
1. Let R = cfw_(x, y, z ) | 0 x, y, z 1 be the solid cube having the
origin as one of its vertices and having its three incident edges
aligned with the positive x, y , and z -axes, respectively. Let
T = cfw_(x, y, z ) | x + y +

LB 220 Calculus III, Quiz 10 Solutions
1. (5 points) Compute the following integral:
F dr,
C
where F = 1 + x2 i+xy 2 j and C is the counter-clockwise oriented boundary of the triangle having vertices (0, 0), (1, 0), and (0, 2).
Let M =
1 + x2 and N = xy 2

Selected Solutions to Homework # 3
1. Matching: Solve exercises # 1 - 12 in section 9.6 of the textbook.
Answers: d, i, a, g, l, e, b, j, k, f, h, c.
2. Solve exercise # 34 in section 9.6.
The graph is a a double circular cone with central axis the z -axi