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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
March 27, 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

4 Multiple Integrals 115
4.1 DoubleIntegrals ...................................... 115

4.2 DoubleIntegralsOverNon-rectangularRegions . . . . . . . . . . . . . . . . . . . . . 125
4.3 DoubleIntegralsinPolarCoordinates .......................... 136

4.4 TripleIntegrals ....................................... 148

4.5 TripleIntegralsinCylindricalCoordinates........................ 155

4.6 TripleIntegralsinSphericalCoordinates ......................... 158

Chapter 4

Multiple Integrals
4.1 Double Integrals
Before starting on double integrals let us do a quick review of what is the de.nition of de.nite integrals for functions of one single variable. For these integrals we are integrating over the interval a . x . b,
b
Z
f(x) dx.
a
Now, when we are going to derive the de.nition of the de.nite integral we .rst think of this as a classical area problem. What is the area problem? Historically, the idea of limits has been used to .nd areas of di.erent shapes. Let us .rst consider a simple example: how the area of a circle can be found by a limiting process?
r
height about same as radius
width approximately = half circumference
We divide the circular disc into pieces of equal size and then rearrange the pieces as in the .gure above. In the limit of increasing number of pieces (i.e., smaller and smaller divisions) the rearrangement will give something closer to a rectangle (imagine it) of which the area can easily be found. Thus the area of the circle is the same as the area of the limiti