Calculus III
Multivariate Calculus
Lecture Notes
Veselin Jungic & Jamie Mulholland
Department of Mathematics
Simon Fraser University
c Jungic/Mulholland, September 25, 2019
License is granted to print this
document for personal/educational use.
,Contents
Contents i
Preface iii
Greek Alphabet v
12 Vectors and the Geometry of Space 1
12.1 Three-Dimensional Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
12.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
12.3 The Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
12.4 The Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
12.5 Equations of Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
12.6 Cylinders and Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
13 Vector Functions 35
13.1 Vector Functions and Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
13.2 Derivatives and Integrals of Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
13.3 Arc Length and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
13.4 Motion in Space: Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
14 Partial Derivatives 57
14.1 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
14.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
14.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
14.4 Tangent Planes and Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
14.5 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
14.6 Directional Derivatives and the Gradient Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 81
14.7 Maximum and Minimum Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
14.8 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
i
,ii CONTENTS
15 Multiple Integrals 101
15.1 Double Integrals over Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
15.2 Double Integrals Over General Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
15.3 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
15.4 Applications of Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
15.5 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
15.6 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
15.7 Triple Integrals in Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
15.8 Triple Integrals in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
15.9 Change of Variables in Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
16 Vector Calculus 147
16.1 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
16.2 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
16.3 Fundamental Theorem for Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
16.4 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
16.5 Curl and Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
16.6 Integral Theorems Summary, and a Look to Calculus IV . . . . . . . . . . . . . . . . . . . . . 170
Bibliography 171
, Preface
This booklet contains our notes for courses Math 251 - Calculus III at Simon Fraser University. Students
are expected to use this booklet during each lecture by follow along with the instructor, filling in the details
in the blanks provided, during the lecture.
Definitions of terms are stated in orange boxes and theorems appear in blue boxes .
Next to some examples you’ll see [link to applet]. The link will take you to an online interactive applet to
accompany the example - just like the ones used by your instructor in the lecture. Clicking the link above
will take you to the following website containing all the applets:
http://www.sfu.ca/ jtmulhol/calculus-applets/html/appletsforcalculus.html
Try it now.
No project such as this can be free from errors and incompleteness. We will be grateful to everyone
who points out any typos, incorrect statements, or sends any other suggestion on how to improve this
manuscript.
Veselin Jungic
Simon Fraser University
Jamie Mulholland
Simon Fraser University
j
September 25, 2019
iii
Multivariate Calculus
Lecture Notes
Veselin Jungic & Jamie Mulholland
Department of Mathematics
Simon Fraser University
c Jungic/Mulholland, September 25, 2019
License is granted to print this
document for personal/educational use.
,Contents
Contents i
Preface iii
Greek Alphabet v
12 Vectors and the Geometry of Space 1
12.1 Three-Dimensional Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
12.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
12.3 The Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
12.4 The Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
12.5 Equations of Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
12.6 Cylinders and Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
13 Vector Functions 35
13.1 Vector Functions and Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
13.2 Derivatives and Integrals of Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
13.3 Arc Length and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
13.4 Motion in Space: Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
14 Partial Derivatives 57
14.1 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
14.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
14.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
14.4 Tangent Planes and Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
14.5 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
14.6 Directional Derivatives and the Gradient Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 81
14.7 Maximum and Minimum Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
14.8 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
i
,ii CONTENTS
15 Multiple Integrals 101
15.1 Double Integrals over Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
15.2 Double Integrals Over General Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
15.3 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
15.4 Applications of Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
15.5 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
15.6 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
15.7 Triple Integrals in Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
15.8 Triple Integrals in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
15.9 Change of Variables in Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
16 Vector Calculus 147
16.1 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
16.2 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
16.3 Fundamental Theorem for Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
16.4 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
16.5 Curl and Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
16.6 Integral Theorems Summary, and a Look to Calculus IV . . . . . . . . . . . . . . . . . . . . . 170
Bibliography 171
, Preface
This booklet contains our notes for courses Math 251 - Calculus III at Simon Fraser University. Students
are expected to use this booklet during each lecture by follow along with the instructor, filling in the details
in the blanks provided, during the lecture.
Definitions of terms are stated in orange boxes and theorems appear in blue boxes .
Next to some examples you’ll see [link to applet]. The link will take you to an online interactive applet to
accompany the example - just like the ones used by your instructor in the lecture. Clicking the link above
will take you to the following website containing all the applets:
http://www.sfu.ca/ jtmulhol/calculus-applets/html/appletsforcalculus.html
Try it now.
No project such as this can be free from errors and incompleteness. We will be grateful to everyone
who points out any typos, incorrect statements, or sends any other suggestion on how to improve this
manuscript.
Veselin Jungic
Simon Fraser University
Jamie Mulholland
Simon Fraser University
j
September 25, 2019
iii
