Multivariable and Complex Calculus
Matthew D. Finn
&
Sanjeeva Balasuriya
,Preface
This textbook has been written for the University of Adelaide course “Multivariable and Complex Calculus”
(MCC), with course codes MATH 2101 and 7101. The course consists of two main parts:
• Multivariable calculus (Chapters 1–4)
• Complex calculus (Chapter 5)
The learning outcomes for each section in the textbook are outlined at the end of the section, to enable
students to review whether they have understood the main points.
Students are urged to consider the fact that lecture recordings will cover the material that is in this text-
book differently, providing a framework for understanding the material, as well as alternative explanations,
additional motivation, and new examples. Therefore, students are urged to first view the recorded lecture
material, and then read the relevant sections of the textbook – this will help read this textbook in context.
(It is well known that simply reading mathematical material directly, without guidance, is not easy!)
The full course consists of all the material in:
• The lectures (presented as video recordings);
• This textbook;
• The tutorials;
• The assignments;
• Any solutions provided;
• Any other resources made available via MyUni or otherwise.
The course is designed such that students are expected to achieve the learning outcomes via actively engaging
with all the above aspects of the course, and not just one source. Information from all of these is examinable,
unless specifically labelled as “optional.”
Students are additionally advised that examinations/tests/assignments will focus on material as covered in
this course in this semester. Since the course material is refreshed each semester to continually update
the course for the modern world, examinations/assignments from previous semesters may or may not be
relevant. Your instructor will provide guidance as to what to expect in examinations and tests. Students
are in general responsible for not just knowing how to follow mathematical procedures, but also developing
both theoretical and conceptual intuition into them, critically understanding what they can and cannot do,
and applying them in (potentially unfamiliar) situations. It is these skills that students will take from this
course into their future studies, and will discover that the course material is relevant across a range of both
theoretical and applied subject areas.
This textbook is collated from classroom material prepared by Matthew D. Finn in 2011. Extensive modi-
fications, rearrangements and additions by Sanjeeva Balasuriya have occurred since 2019, to arrange in the
format of a textbook. Please report any typographic errors discovered to your instructor for correction.
January 29, 2021
1
,Contents
Preface 1
1 Multivariable functions 4
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Sets and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Suffix notation for vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Scalar- and vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Limits and continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Open and closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Differentiating multivariable functions 20
2.1 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.1 Scalar-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.2 Vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.3 Properties of derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.4 Higher-derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Taylor’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2
, CONTENTS 3
2.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 The divergence and the curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6 Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Integrating multivariable functions 56
3.1 Integration over areas (double integrals) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Integration over volumes (triple integrals) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Change-of-variables in multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4 Integrals along curves (line or path integrals) . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Integrals over surfaces (surface and flux integrals) . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Fundamental theorems of multivariable calculus 82
4.1 Green’s theorem in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2 Stokes’ theorem in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3 Gauss’ (divergence) theorem in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Complex calculus 97
5.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 Differentiating complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4 Integrating complex functions (contour integration) . . . . . . . . . . . . . . . . . . . . . . . . 111
5.5 Laurent series and the residue theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Matthew D. Finn
&
Sanjeeva Balasuriya
,Preface
This textbook has been written for the University of Adelaide course “Multivariable and Complex Calculus”
(MCC), with course codes MATH 2101 and 7101. The course consists of two main parts:
• Multivariable calculus (Chapters 1–4)
• Complex calculus (Chapter 5)
The learning outcomes for each section in the textbook are outlined at the end of the section, to enable
students to review whether they have understood the main points.
Students are urged to consider the fact that lecture recordings will cover the material that is in this text-
book differently, providing a framework for understanding the material, as well as alternative explanations,
additional motivation, and new examples. Therefore, students are urged to first view the recorded lecture
material, and then read the relevant sections of the textbook – this will help read this textbook in context.
(It is well known that simply reading mathematical material directly, without guidance, is not easy!)
The full course consists of all the material in:
• The lectures (presented as video recordings);
• This textbook;
• The tutorials;
• The assignments;
• Any solutions provided;
• Any other resources made available via MyUni or otherwise.
The course is designed such that students are expected to achieve the learning outcomes via actively engaging
with all the above aspects of the course, and not just one source. Information from all of these is examinable,
unless specifically labelled as “optional.”
Students are additionally advised that examinations/tests/assignments will focus on material as covered in
this course in this semester. Since the course material is refreshed each semester to continually update
the course for the modern world, examinations/assignments from previous semesters may or may not be
relevant. Your instructor will provide guidance as to what to expect in examinations and tests. Students
are in general responsible for not just knowing how to follow mathematical procedures, but also developing
both theoretical and conceptual intuition into them, critically understanding what they can and cannot do,
and applying them in (potentially unfamiliar) situations. It is these skills that students will take from this
course into their future studies, and will discover that the course material is relevant across a range of both
theoretical and applied subject areas.
This textbook is collated from classroom material prepared by Matthew D. Finn in 2011. Extensive modi-
fications, rearrangements and additions by Sanjeeva Balasuriya have occurred since 2019, to arrange in the
format of a textbook. Please report any typographic errors discovered to your instructor for correction.
January 29, 2021
1
,Contents
Preface 1
1 Multivariable functions 4
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Sets and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Suffix notation for vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Scalar- and vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Limits and continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Open and closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Differentiating multivariable functions 20
2.1 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.1 Scalar-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.2 Vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.3 Properties of derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.4 Higher-derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Taylor’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2
, CONTENTS 3
2.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 The divergence and the curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6 Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Integrating multivariable functions 56
3.1 Integration over areas (double integrals) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Integration over volumes (triple integrals) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Change-of-variables in multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4 Integrals along curves (line or path integrals) . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Integrals over surfaces (surface and flux integrals) . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Fundamental theorems of multivariable calculus 82
4.1 Green’s theorem in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2 Stokes’ theorem in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3 Gauss’ (divergence) theorem in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Complex calculus 97
5.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 Differentiating complex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4 Integrating complex functions (contour integration) . . . . . . . . . . . . . . . . . . . . . . . . 111
5.5 Laurent series and the residue theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
