Calculus III
Preface
Here are my online notes for my Calculus III course that I teach here at Lamar University.
Despite the fact that these are my class notes, they should be accessible to anyone wanting to
learn Calculus III or needing a refresher in som
Parametric Representations
of
Plane Curves
y
x = t2-1
y = t3 -t
x
A plane curve is a 2-dimensional curve given by
where f and g are continuous functions on the interval I, [a,b].
t is the parameter.
Q
P
(x(b), y(b)
(x(a), y(a)
terminal endpoint
initial en
Cartesian Coordinates
in
3-Space
A point in 3-space is given by an ordered triple (x,y,z).
z
y
x
z
z
Plot (2,3,4)
y
y
x
x
Plot (-1,4,-3)
1
Distance Formula
d2 = (x)2 + (y)2 + (z)2
z
y
x
EX 1 Show that these points are vertices of an equilateral triangle.
A Geometric and
Algebraic Approach
to
Vectors
u
v
u+v=v+u
u
v
VECTORS (Geometric Approach)
Scalar
Vector
Magnitude
Direction
if they have the same magnitude and direction.
zero vector
and
scalar multiple of
, where c is a real number,
means we have a vec
Cylindrical and Spherical Coordinates
We can describe a point, P, in three different ways.
Cartesian
Cylindrical
Spherical
Cylindrical Coordinates
x = r cos
r = x2 + y2
y = r sin
tan = y/x
z=z
z=z
Spherical Coordinates
x = sincos
= x2 + y2 + z2
y = sinsi
Lines and Tangent Lines
in
3-Space
A 3-D curve can be given parametrically by x = f(t), y = g(t) and z = h(t)
where t is on some interval I and f, g, and h are all continuous on I.
We could specify the curve by the position vector
.
Given a point P0, dete
Surfaces in Three-Space
Quick Review of the Conic Sections
a) Parabola
b) Ellipse
c) Hyperbola
=1
1
Surfaces in Three-Space
The graph of a 3-variable equation which can be written in the form
F(x,y,z) = 0 or sometimes z = f(x,y) (if you can solve for z) i
The Cross Product
The Cross Product
where
and
.
1
EX 1 If
=3,3,1and
find
=-2,-1,0and
= -2,-3,-1,
.
Theorem A
Let
and
be 3-D vectors and is the angle between them.
Then
1)
and
2)
3)
form a right-handed triple.
2
Theorem B
Two 3-D vectors,
and
are parallel
Course Title: Analytic Geometry & Calculus III
Class Hours: 4
Course No. MAT 233
Laboratory Hours: 0
Department Head Approval: _
Maria DeLucia, Ph.D.
Credit Hours: 4
Date: 2007-2008
Dean Approval: _
Reginald Luke, Ph.D.
Prerequisite:
MAT 132 (Analytic Geo
Staffing Team Project
Your team will conduct a job analysis and develop a selection interview for a specific job Your
team will also define your overall selection approach to ensure the selection of the best applicant.
Job Analysis
You will need to conduc
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Vector-Valued Functions
and
Curvilinear Motion
A vector-valued function associates a vector output,
,
to a scalar input.
i.e
(in 2D)
where f and g are real-valued functions of t
or
(in 3D).
Definition
means that for every
corresponding
such that
there is