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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 28, 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

3 Partial Derivatives 49
3.1 FunctionsofSeveralVariables............................... 49

3.2 LimitsandContinuity ................................... 53

3.3 PartialDerivatives ..................................... 58

3.4 TheChainRule....................................... 73

3.5 DirectionalDerivatives ................................... 81

3.6 ApplicationsofPartialDerivatives ............................ 89

Chapter 3
Partial Derivatives
The notation y = f(x) is used to indicate that the variable y depends on the single independent variable x, that is, that y is a function of x. In fact, many functions depend on more than one independent variable. For instance, the volume of a circular cone is a function V = (1/3)r2h of its radius and its height, so it is a function V (r, h) of two variables. In this chapter we extend the basic ideas of single variable calculus to functions of several variables.
The calculus of several variables is basically single variable calculus applied to several variables once at a time. When we hold all but one of the independent variables of a function constant and di.erentiate with respect to that one variable, we get a partial derivative. Section 3.3 (page 58) will show how partial derivatives are de.ned and interpreted geometrically, and how to calculate them by applying the rules for di.erentiating functions of a single variable. Despite the fact that this chapter is about derivatives we would like to .rst develop the fundamentals and to introduce the basic concepts on limits and continuity of functions of several variables.
3.1 Functions of Several Variables
In this beginning section we .rst de.ne functions of more than one independent variable and discuss the their geometric representations.
Real-valued functions of several independent variables are de.ned similarly to functions in the single variable case. By analogy with the corresponding de.nition for functions of single v