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MATH 100 Spring 2006-07

Introduction to Multivariable Calculus

Lecture Notes
Dr. Tony Yee
Department of Mathematics The Hong Kong University of Science and Technology
February 8, 2007

Contents
Table of Contents iii
1 Vectors and Geometry of Space 1
1.1 Three-DimensionalCoordinateSystems ......................... 1

1.2 Vectors ........................................... 5

1.3 TheDotProduct ...................................... 10

1.4 TheCrossProduct ..................................... 13

1.5 EquationsofLines ..................................... 18

1.6 EquationsofPlanes .................................... 22

1.7 QuadricSurfaces ...................................... 29

2 Vector-Valued Functions 33
2.1 VectorFunctions ...................................... 33

2.2 CalculuswithVectorFunctions .............................. 39

2.3 Tangent,NormalandBinormalVectors ......................... 42

2.4 ArcLengthinSpace .................................... 46

Chapter 2
Vector-Valued Functions
2.1 Vector Functions
R3
So far we have seen some examples of graphing surfaces in . However, as we saw with lines, not every
R3
graph in needs to be a surface. We can graph curves (sometimes called space curves) that are
three-dimensional as well. To do this we used vector-valued functions or shortly vector functions. Note that we can also use vector functions to represent surfaces as well as we will see at the end of this section. With that being said however we will spend most of this section talking about curves instead of surfaces. The vector form of the equation of a line is a good example of a vector function. For instance,
r(t)= r0 + tv.
Vector functions take real numbers (or scalars) as arguments, t in this case, and return vectors that are the position vectors for points on the curve (or surface).
The general form of a three-dimensional vector function for a curve is
r(t)= .f(t),g(t),h(t)., or r(t)= f(t) i + g(t) j + h(t) k,
where f(t), g(t), and h(t) are sometimes called the component functions. In the following .gure the
.
moving point P (x, y, z)=(f(t),g(t),h(t)) (with r(t)= OP being its position vector) makes up the curve in space. We say that the equations x = f(t), y = g(t), z = h(t) parametrize the curve. Each .xed value of t will determine one point on the curve. Here r(t) is a vector function whereas the components of r are scalar functions of t.
z
The Curve
P (f(t),g(t),h(t))
r
O y
x
2. Vector-Valued Functions
The domain of a vector function is the set of all ts for which all the component functions are de.ned. For now, the domains will be intervals of real numbers resulting in a space curve. Later, in Chapter 5, the domains will be regions in the plane. Vector functions will then represent surfaces in space. Vector functions on a domain in the plane or space also give rise to vector .elds, which are important to the study of the .ow of a .uid