Math 321
Vector and Complex Calculus
for the Physical Sciences
©Fabian Wale e
Department of Mathematics &
Department of Engineering Physics
University of Wisconsin, Madison
January 25, 20182 ©F. Wale e, Math 321,
...
Math 321
Vector and Complex Calculus
for the Physical Sciences
©Fabian Wale e
Department of Mathematics &
Department of Engineering Physics
University of Wisconsin, Madison
January 25, 20182 ©F. Wale e, Math 321, January 25, 2018
TERMS OF USE:
These notes and other materials (homework, exams, solutions) are provided for
the private use of current and past Math 321 students at UW-Madison. You should
not redistribute any of these materials or post them onto external websites such as
coursehero.com, or any other such websites.
We are providing many ways to get help on the material. In addition to these comprehensive notes, we o er interactive lectures, discussion sections and o ce hours by
faculty and TAs as well as interactive course websites such as piazza.com, where
students can ask and answer questions, including anonymously if they prefer.Preface
These notes have been developed to ll the gap between the mathematics covered in
standard multivariable calculus courses and the vector and complex calculus needed
for intermediate and advanced undergraduate courses in the physical and engineering
sciences. Emphasis is placed on geometric understanding of the fundamental concepts
and connections with physical world experiences and applications. Vector, index and
matrix notations are discussed since each have their own merits and limitations and
all three are used interchangeably in applications.
The mathematical concepts and tools covered here are fundamental to sciences
such as mechanics, electromagnetism, uid dynamics, aerodynamics, transport phenomena, ight dynamics, astrodynamics, continuum mechanics, elasticity, plasma physics,
geophysical and astrophysical uid dynamics, for example, as well as computer graphics
that requires a deeper understanding of the mathematical modeling of curves, surfaces
and volumes. It is common for instructors of courses in those areas to spend 1/3 or
more of their time quickly ‘reviewing’ the requisite vector and complex calculus that
is needed for adequate presentation and understanding of the material. This creates
excessive redundancy at an introductory level that is frustrating and ine cient for
students and instructors alike.
The material is presented at an intermediate to advanced undergraduate level.
Students are expected to have mastered basic geometry, trigonometry, algebra and
calculus including multi-variable calculus (partial derivatives, multiple integrals). Familiarity with di erential equations and linear algebra is recommended but not perhaps not required. Knowledge of introductory physics is assumed. The concepts are
motivated, derived and justied carefully but the approach is intended to be more
‘intuitive’ than ‘rigorous,’ although those terms are subjective. The derivations and
proofs should be studied and understood as they lead to deeper understanding of the
concepts. Solved examples are provided but not too many as to drown out the fundamental concepts. The concepts and examples should be studied thoroughly before
attempting the exercises. Exercises appear after most subsections to encourage the
productive cycle of studying a particular concept then practicing its applications in
non-trivial problems and going back to studying the concept more deeply. Learning
is an iterative process.
Acknowledgements: Thanks to Prof. Jean-Luc Thi eault and Drs. Anakewit ‘Tete’
Boonkasame and Alfredo Wetzel for using and commenting on these notes. Skull surface image courtesy of https://doc.cgal.org/latest/Surface_mesher/
index.html.
34 ©F. Wale e, Math 321, January 25, 2018Contents
1 Vector Geometry and Algebra 9
1 Find your bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1 Magnitude and Direction . . . . . . . . . . . . . . . . . . . . . 9
1.2 Vectors on a plane . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Vectors in space . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Addition and scaling of vectors . . . . . . . . . . . . . . . . . 14
2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Abstract Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The vector space Rn . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Bases and Components . . . . . . . . . . . . . . . . . . . . . . 18
3 Points and Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1 Position vector . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . 22
3.3 Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Medians of a triangle . . . . . . . . . . . . . . . . . . . . . . . 24
4 Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1 Geometry and algebra of dot product . . . . . . . . . . . . . . 26
4.2 Orthonormal bases . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Dot product and norm in Rn (Optional) . . . . . . . . . . . . 33
5 Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1 Geometry and algebra of cross product . . . . . . . . . . . . . 35
5.2 Double cross product (‘Triple vector product’) . . . . . . . . . 37
5.3 Orientation of Bases . . . . . . . . . . . . . . . . . . . . . . . 38
6 Index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.1 Levi-Civita symbol . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2 Einstein summation convention . . . . . . . . . . . . . . . . . 44
7 Mixed product and Determinant . . . . . . . . . . . . . . . . . . . . . 48
8 Points, Lines, Planes, etc. . . . . . . . . . . . . . . . . . . . . . . . . . 53
9 Orthogonal Transformations and Matrices . . . . . . . . . . . . . . . 56
9.1 Change of cartesian basis in 2D . . . . . . . . . . . . . . . . . 56
9.2 Change of cartesian basis in 3D . . . . . . . . . . . . . . . . . 57
9.3 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
9.4 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
9.5 Determinant of a matrix (Optional) . . . . . . . . . . . . . . . 69
56 ©F. Wale e, Math 321, January 25, 2018
9.6 Three views of Ax = b (Optional) . . . . . . . . . . . . . . . 69
9.7 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
9.8 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 72
2 Vector Calculus 73
1 Vector functions and their derivatives . . . . . . . . . . . . . . . . . . 73
2 Newtonian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.1 Motion of a single particle . . . . . . . . . . . . . . . . . . . . 79
2.2 Motion of a system of particles (optional) . . . . . . . . . . . 83
2.3 Motion of a rigid body (optional) . . . . . . . . . . . . . . . . 84
3 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.1 Elementary curves . . . . . . . . . . . . . . . . . . . . . . . . 88
3.2 Speeding through Curves . . . . . . . . . . . . . . . . . . . . 90
3.3 Integrals over curves . . . . . . . . . . . . . . . . . . . . . . . 92
4 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.1 Parameterizations . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2 Surface integrals . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3 Curves on surfaces . . . . . . . . . . . . . . . . . . . . . . . . 105
5 Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6 Maps, curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . 108
7 Change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8 Grad, div, curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.1 Scalar elds and iso-contours . . . . . . . . . . . . . . . . . . 117
8.2 Geometric concept of the Gradient . . . . . . . . . . . . . . . 117
8.3 Directional derivative, gradient and the r operator . . . . . . 119
8.4 Div and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.5 Vector identities . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.6 Grad, Div, Curl in cylindrical and spherical coordinates . . . . 125
9 Inverting Grad, Div, Curl (optional) . . . . . . . . . . . . . . . . . . . 130
10 Fundamental theorems of vector calculus . . . . . . . . . . . . . . . . 131
10.1 Integration in R2 and R3 . . . . . . . . . . . . . . . . . . . . . 131
10.2 Fundamental theorem of Calculus . . . . . . . . . . . . . . . . 132
10.3 Fundamental theorem in R2 . . . . . . . . . . . . . . . . . . . 132
10.4 Green and Stokes’ theorems . . . . . . . . . . . . . . . . . . . 134
10.5 Divergence form of Green’s theorem . . . . . . . . . . . . . . 137
10.6 Gauss’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 138
10.7 Other forms of the fundamental theorem in 3D . . . . . . . . 140
3 Complex Calculus 143
1 Complex Numbers and Elementary functions . . . . . . . . . . . . . . 143
1.1 Complex Algebra and Geometry . . . . . . . . . . . . . . . . 143
1.2 Limits and Derivatives . . . . . . . . . . . . . . . . . . . . . . 148
1.3 Geometric sums and series . . . . . . . . . . . . . . . . . . . . 149
1.4 Ratio test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
1.5 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
1.6 Complex transcendentals . . . . . . . . . . . . . . . . . . . . 154©F. Wale e, Math 321, January 25, 2018 7
1.7 Polar representation . . . . . . . . . . . . . . . . . . . . . . . 158
1.8 Logs and complex powers . . . . . . . . . . . . . . . . . . . . 159
2 Functions of a complex variable . . . . . . . . . . . . . . . . . . . . . 164
2.1 Visualization of complex functions . . . . . . . . . . . . . . . 164
2.2 Cauchy-Riemann equations . . . . . . . . . . . . . . . . . . . 165
2.3 Geometry of Cauchy-Riemann, Conformal Mapping . . . . . 170
2.4 Conformal mapping examples . . . . . . . . . . . . . . . . . . 174
2.5 Joukowski Map . . . . . . . . . . . . . . . . . . . . . . . . . . 176
3 Integration of Complex Functions . . . . . . . . . . . . . . . . . . . . 179
3.1 Complex integrals are path integrals . . . . . . . . . . . . . . 179
3.2 Cauchy’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 181
3.3 Poles and Residues . . . . . . . . . . . . . . . . . . . . . . . . 181
3.4 Cauchy’s formula . . . . . . . . . . . . . . . . . . . . . . . . . 186
4 Real examples of complex integration . . . . . . . . . . . . . . . . . . 1908 ©F. Wale e, Math 321, January 25, 2018Chapter 1
Vector Geometry and Algebra
1 Find your bearings
What is a vector? In calculus, you may have dened vectors as lists of numbers such
as (a1; a2) or (a1; a2; a3). Mathematicians favor that approach nowadays because
it readily generalizes to n dimensional space (a1; a2; : : : ; an) and reduces geometry
to arithmetic. But in physics, and in geometry before that, you encountered vectors
as quantities with both magnitude and direction such as displacements, velocities and
forces. The geometric point of view emphasizes the invariance of these quantities with
respect to changes of the system of coordinates.
1.1 Magnitude and Direction
The prototypical vectors are the displacements in two or three dimensional space such
as the displacement of a boat on the surface of a lake, or a drone ying over a prairie.
We write
a = a a^ (1)
for a vector a of magnitude a in the direction a^.
N
α
a a
Fig. 1.1: Vector a with
magnitude a, heading α
In navigation, the direction a^ is specied by the heading (or azimuth) α which
is the clockwise angle from the Northern direction. The magnitude a is specied in
appropriate physical units such as meters for a displacement, or meters/second for a
velocity. Magnitude does not have to come before direction, ‘1km Northeast’ is the
same as ‘Northeast 1km,’
a a^ = a^ a:
Vectors are denoted in boldface type such as a, b, u, v, . . . , usually lower-case but
often upper-case also in Physics, for example for the magnetic eld B, electric eld
E or resultant force F . The magnitude of vector a is written in regular type a or with
vertical bars jaj
jaj = jaa^j = a ≥ 0: (2
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