Introduction to Mathmatical Physics
Edition: 1
Copyright: 2026
Pages: 506
Edition: 1
Copyright: 2026
Pages: 506
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This textbook introduces multivariable calculus and its applications in physics and engineering. It is intended primarily for second-year engineering and physics students and assumes a foundational background in linear algebra together with a strong understanding of single-variable calculus, typically acquired through a first course in linear algebra and two semesters of first-year calculus. The text is designed to provide the level of understanding required for physics and engineering applications in a concise and accessible manner. As such, the emphasis is placed on intuition, geometric insight, practical interpretation, and computational proficiency rather than on deep theoretical proofs or technical formalities.
Definitions and theorems are introduced in a practically motivated way, with a focus on developing conceptual understanding and problem-solving skills relevant to scientific and engineering contexts. While the primary purpose of this text is to serve as a course textbook for engineering and physics students, it was also written with a secondary goal in mind: to complement more rigorous and proof-oriented texts such as Calculus on Manifolds by Spivak and Analysis on Manifolds by Munkres. In contrast to the highly
abstract and theoretical presentation found in such works, this textbook emphasizes the computational, geometric, and applied aspects of the subject, helping students connect rigorous mathematical ideas with practical applications in science and engineering.
About this Textbook
About the Authors
Acknowledgements
Chapter 1 Common 3D Geometries and Their Properties
1.1 Lines in Space
1.2 Equations of Planes
1.3 Parallel and Orthogonal Planes
1.4 Cylinders and Traces
1.5 Quadric Surfaces and Traces
1.6 Exercises
Chapter 2 Multivariable Functions
2.1 Scalar Fields
2.2 Curves
2.3 Coordinate Transformations
2.4 Vector Fields
2.5 Understanding Vectors, Domains, and Codomains
2.6 Exercises
Chapter 3 Limits and Continuity
3.1 Limits
3.2 Continuity
3.3 Exercises
Chapter 4 Differentiation in Multivariables
4.1 Partial and Directional Derivatives
4.2 Tangent Plane
4.3 Differentiability
4.4 Interpretation of the Gradient
4.5 Derivative of Vector-Valued Functions
4.6 Chain Rule
4.7 Exercises
Chapter 5 Applications of Derivatives
5.1 Taylor Series
5.2 Directional Derivative Applied to a Kinematic Problem
5.2.1 Problem Description
5.2.2 Problem Statement
5.2.3 Solution
5.3 Small-Signal Analysis
5.3.1 Small Signal Model for a Single Input and Single Output
5.3.2 Small Signal Model for the Multivariable Case Given a System Model
5.3.3 System Model
5.3.4 Example Systems Consisting of Two State Variables, Two Input Variables, and Two Output Variables
5.3.5 Small-Signal Modeling Procedure
5.3.6 Example Problems
5.4 Exercises
Chapter 6 Integrals in two dimensions
6.1 Fubini’s Theorem for Rectangles
6.2 Fubini’s Theorem for Simple Regions
6.3 Swapping the Order of Integration
6.4 Exercises
Chapter 7 Integrals in three dimensions
7.1 Fubini’s Theorem Over Rectangular Boxes
7.2 Fubini’s Theorem Over Simple Regions
7.3 Swapping the Order of Integration
7.4 Exercises
Chapter 8 Change of Variables
8.1 Choosing an Appropriate Change of Variables
8.2 Polar Coordinates
8.3 Change of Variables in Three Dimensions
8.4 Cylindrical Coordinates
8.5 Spherical Coordinates
8.6 Exercises
Chapter 9 Calculus of Curves
9.1 Introduction to Curves
9.2 Integrating Scalar Functions Over Curves
9.3 Circulation Integrals
9.4 Flux Integrals in Two Dimensions
9.5 Exercises
Chapter 10 Calculus of Surfaces
10.1 Integrating Scalar Functions Over Surfaces
10.2 Flux Integrals in Three Dimensions
10.3 Exercises
Chapter 11 The Fundamental Theorems of Multivariable Calculus
11.1 The Divergence and Curl Operators
11.2 The Fundamental Theorem for Line Integrals
11.3 Divergence Theorem
11.4 Stoke’s Theorem
11.5 Exercises
Chapter 12 Vector Operators, Vector Operator Identities, and Helmholtz Decomposition Theorem
12.1 Vector Operators
12.2 Vector Operator Identities
12.3 Helmholtz Decomposition Theorem
12.4 Exercises
Chapter 13 Vector Calculus Application Tools
13.1 Computing a Vector Field F When ∇ × F and ∇ · F Are Given
13.2 Modeling and Applying Dirac–Distribution Functions
13.3 Vectors and Coordinate Transformations
13.4 Superposition for Vector Fields
13.5 Exercises
Chapter 14 Computing Fields Under Conditions of Symmetry for an Irrotational Field
14.1 Planar
14.2 Cylindrical
14.3 Spherical
14.4 Complicated Geometries for Which Superposition Can be Applied
14.5 Exercises
Chapter 15 Computing Fields Under Conditions of Symmetry for a Source-Free Field
15.1 Planar
15.2 Cylindrical (Axial Current Flux Density)
15.3 Infinite Axial Helix, or Circumferential Current Flux Density Around an Infinitely Long Cylinder or Cylindrical Shell
15.4 Toroidal Helix or Circumferential Sheet Current Flux Density Around the Torus
15.5 Complicated Geometries for Which Superposition Can be Applied
15.6 Exercises
Appendix A Transformation of Generalized Quadric Equation to a Canonical Form
Appendix B Divergence and Stoke’s Theorem Representation in 2D and Special Cases
B.1 Divergence Theorem in 2D
B.2 Stoke’s Theorem in 2D
B.3 Applying the Divergence and Stoke’s Theorem for Special Cases
B.4 Contours Surfaces and Regions
B.5 Surfaces and Regions in 3D
Solutions Manual for Chapters 5, 12, 13, 14 and 15
Francis Dawson (S’86-M’87) received the B.Sc. degree in physics and the B.A.Sc., M.A.Sc., and Ph.D. degrees in electrical engineering from the University of Toronto in 1978, 1982, 1985, and 1988, respectively. He worked as a process control engineer in the pulp and paper, rubber, and textile industries during the period 1978 to 1980. From 1982 to 1984, he acted as a consultant on various projects. Development areas included high-frequency link power supplies, power supplies for specialized applications, and high current protection circuits. Since 1988 he has been with the Department of Electrical and Computer Engineering, University of Toronto, where he is engaged in teaching and research. His areas of research interest include static power converters and their applications, signal processing in power engineering applications, energy storage systems, and device or process modeling. He has also participated as a consultant or project leader in several industrial projects. Dr. Dawson is a member of the Association of Professional Engineers of Ontario and is an IEEE Fellow.
Armanpreet Pannu received his B.Sc. degree in mathematics and physics and his M.Sc. degree in mathematics from the University of Toronto in 2017 and 2018, respectively. Following this, he began doctoral studies in the Department of Mathematics in the area of quantum information theory before transitioning to the Department of Electrical and Computer Engineering, where he completed his Ph.D. in theoretical quantum photonics in 2025. His doctoral research focused on mathematical and analytic methods in quantum photonics and quantum information, with an emphasis on quantum-enhanced sensing using discrete-variable entangled states. Dr. Pannu has maintained a strong interest in bridging the gap between pure mathematics and applications in physics and engineering through both his research and teaching. He has taught a wide range of undergraduate mathematics and engineering courses at the University of Toronto, including multiple offerings of multivariable calculus ranging from rigorous proof-based courses for mathematics students to more applied versions tailored toward engineering and the life sciences. This broad teaching experience and emphasis on intuitive yet rigorous mathematical understanding served as a motivation for this textbook