Calculus Illustrated. Volume 4: Calculus in Higher Dimensions

510 Pages·64.836 MB·English

by  Peter Saveliev

1
To the student... 2
To the student
Mathematics is a science. Just as the rest of the scientists, mathematicians are trying to understand how
the Universe operates and discover its laws. When successful, they write these laws as short statements
called (cid:16)theorems(cid:17). In order to present these laws conclusively and precisely, a dictionary of the new concepts
is also developed; its entries are called (cid:16)de(cid:28)nitions(cid:17). These two make up the most important part of any
mathematics book.
This is how de(cid:28)nitions, theorems, and some other items are used as building blocks of the scienti(cid:28)c theory
we present in this text.
Every new concept is introduced with utmost speci(cid:28)city.
De(cid:28)nition 0.0.1: square root
Suppose a is a positive number. Then the square root of a is a positive number
x, such that x2 = a.
The term being introduced is given in italics. The de(cid:28)nitions are then constantly referred to throughout
the text.
New symbolism may also be introduced.
Square root
√
a
Consequently, the notation is freely used throughout the text.
We may consider a speci(cid:28)c instance of a new concept either before or after it is explicitly de(cid:28)ned.
Example 0.0.2: length of diagonal
What is the length of the diagonal of a 1×1 square? The square is made of two right triangles and the
diagonal is their shared hypotenuse. Let’s call it a. Then, by the Pythagorean Theorem, the square of
a is 12 +12 = 2. Consequently, we have:
a2 = 2.
√
We immediately see the need for the square root! The length is, therefore, a = 2.
You can skip some of the examples without violating the (cid:29)ow of ideas, at your own risk.
All new material is followed by a few little tasks, or questions, like this.
Exercise 0.0.3
Find the height of an equilateral triangle the length of the side of which is 1.
The exercises are to be attempted (or at least considered) immediately.
Most of the in-text exercises are not elaborate. They aren’t, however, entirely routine as they require
understanding of, at least, the concepts that have just been introduced. Additional exercise sets are placed
intheappendixaswellasatthebook’swebsite: calculus123.com. Donotstartyourstudywiththeexercises!
Keep in mind that the exercises are meant to test (cid:21) indirectly and imperfectly (cid:21) how well the concepts have
been learned.
There are sometimes words of caution about common mistakes made by the students.
To the student... 3
Warning!
In spite of the fact that (−1)2 = 1, there is only
√
one square root of 1, 1 = 1.
The most important facts about the new concepts are put forward in the following manner.
Theorem 0.0.4: Product of Roots
For any two positive numbers a and b, we have the following identity:
√ √ √
a· b = a·b
The theorems are constantly referred to throughout the text.
As you can see, theorems may contain formulas; a theorem supplies limitations on the applicability of the
formula it contains. Furthermore, every formula is a part of a theorem, and using the former without
knowing the latter is perilous.
There is no need to memorize de(cid:28)nitions or theorems (and formulas), initially. With enough time spent with
the material, the main ones will eventually become familiar as they continue to reappear in the text. Watch
for words (cid:16)important(cid:17), (cid:16)crucial(cid:17), etc. Those new concepts that do not reappear in this text are likely to be
seen in the next mathematics book that you read. You need to, however, be aware of all of the de(cid:28)nitions
and theorems and be able to (cid:28)nd the right one when necessary.
Often, but not always, a theorem is followed by a thorough argument as a justi(cid:28)cation.
Proof.
√ √
Suppose A = a and B = b. Then, according to the de(cid:28)nition, we have the following:
a = A2 and b = B2 .
Therefore, we have:
a·b = A2 ·B2 = A·A·B ·B = (A·B)·(A·B) = (AB)2 .
√
Hence, ab = A·B, again according to the de(cid:28)nition.
Some proofs can be skipped at (cid:28)rst reading.
Its highly detailed exposition makes the book a good choice for self-study. If this is your case, these are my
suggestions.
While reading the book, try to make sure that you understand new concepts and ideas. Keep in mind,
however, that some are more important that others; they are marked accordingly. Come back (or jump
forward) as needed. Contemplate. Find other sources if necessary. You should not turn to the exercise sets
until you have become comfortable with the material.
What to do about exercises when solutions aren’t provided? First, use the examples. Many of them contain
a problem (cid:21) with a solution. Try to solve the problem (cid:21) before or after reading the solution. You can also
(cid:28)nd exercises online or make up your own problems and solve them!
I strongly suggest that your solution should be thoroughly written. You should write in complete sentences,
including all the algebra. For example, you should appreciate the di(cid:27)erence between these two:
1+1 1+1
Wrong: Right:
2 = 2
To the student... 4
The latter reads (cid:16)one added to one is two(cid:17), while the former cannot be read. You should also justify all your
steps and conclusions, including all the algebra. For example, you should appreciate the di(cid:27)erence between
these two:
2x = 4 2x = 4; therefore,
Wrong: Right:
x = 2 x = 2.
The standards of thoroughness are provided by the examples in the book.
Next, your solution should be thoroughly read. This is the time for self-criticism: Look for errors and weak
spots. It should be re-read and then rewritten. Once you are convinced that the solution is correct and the
presentation is solid, you may show it to a knowledgeable person for a once-over.
Next, you may turn to modeling projects. Spreadsheets (Microsoft Excel or similar) are chosen to be used
for graphing and modeling. One can achieve as good results with packages speci(cid:28)cally designed for these
purposes, but spreadsheets provide a tool with a wider scope of applications. Programming is another
option.
Good luck!
August 8, 2020
To the teacher 5
To the teacher
The bulk of the material in the book comes from my lecture notes.
There is little emphasis on closed-form computations and algebraic manipulations. I do think that a person
who has never integrated by hand (or di(cid:27)erentiated, or applied the quadratic formula, etc.) cannot possibly
understand integration (or di(cid:27)erentiation, or quadratic functions, etc.). However, a large proportion of time
and e(cid:27)ort can and should be directed toward:
• understanding of the concepts and
• modeling in realistic settings.
The challenge of this approach is that it requires more abstraction rather than less.
Visualization is the main tool used to deal with this challenge. Illustrations are provided for every concept,
big or small. The pictures that come out are sometimes very precise but sometimes serve as mere metaphors
for the concepts they illustrate. The hope is that they will serve as visual (cid:16)anchors(cid:17) in addition to the words
and formulas.
It is unlikely that a person who has never plotted the graph of a function by hand can understand graphs
or functions. However, what if we want to plot more than just a few points in order to visualize curves,
surfaces, vector (cid:28)elds, etc.? Spreadsheets were chosen over graphic calculators for visualization purposes
because they represent the shortest step away from pen and paper. Indeed, the data is plotted in the
simplest manner possible: one cell - one number - one point on the graph. For more advanced tasks such as
modeling, spreadsheets were chosen over other software and programming options for their wide availability
and, above all, their simplicity. Nine out of ten, the spreadsheet shown was initially created from scratch in
front of the students who were later able to follow my footsteps and create their own.
About the tests. The book isn’t designed to prepare the student for some preexisting exam; on the contrary,
assignments should be based on what has been learned. The students’ understanding of the concepts needs
to be tested but, most of the time, this can be done only indirectly. Therefore, a certain share of routine,
mechanical problems is inevitable. Nonetheless, no topic deserves more attention just because it’s likely to
be on the test.
If at all possible, don’t make the students memorize formulas.
In the order of topics, the main di(cid:27)erence from a typical calculus textbook is that sequences come before
everything else. The reasons are the following:
• Sequences are the simplest kind of functions.
• Limits of sequences are simpler than limits of general functions (including the ones at in(cid:28)nity).
• The sigma notation, the Riemann sums, and the Riemann integral make more sense to a student with
a solid background in sequences.
• A quick transition from sequences to series often leads to confusion between the two.
• Sequences are needed for modeling, which should start as early as possible.
From the discrete to the continuous 6
From the discrete to the continuous
It’s no secret that a vast majority of calculus students will never use what they have learned. Poor career
choices aside, a former calculus student is often unable to recognize the mathematics that is supposed to
surround him. Why does this happen?
Calculus is the science of change. From the very beginning, its peculiar challenge has been to study and
measure continuous change: curves and motion along curves. These curves and this motion are represented
by formulas. Skillful manipulation of those formulas is what solves calculus problems. For over 300 years,
this approach has been extremely successful in sciences and engineering. The successes are well-known:
projectile motion, planetary motion, (cid:29)ow of liquids, heat transfer, wave propagation, etc. Teaching calculus
follows this approach: An overwhelming majority of what the student does is manipulation of formulas on
a piece of paper. But this means that all the problems the student faces were (or could have been) solved
in the 18th or 19th centuries!
This isn’t good enough anymore. What has changed since then? The computers have appeared, of course,
and computers don’t manipulate formulas. They don’t help with solving (cid:21) in the traditional sense of
the word (cid:21) those problems from the past centuries. Instead of continuous, computers excel at handling
incremental processes, and instead of formulas they are great at managing discrete (digital) data. To utilize
these advantages, scientists (cid:16)discretize(cid:17) the results of calculus and create algorithms that manipulate the
digital data. The solutions are approximate but the applicability is unlimited. Since the 20th century,
this approach has been extremely successful in sciences and engineering: aerodynamics (airplane and car
design), sound and image processing, space exploration, structure of the atom and the universe, etc. The
approach is also circuitous: Every concept in calculus starts (cid:21) often implicitly (cid:21) as a discrete approximation
of a continuous phenomenon!
Calculus is the science of change, both incremental and continuous. The former part (cid:21) the so-called discrete
calculus (cid:21) may be seen as the study of incremental phenomena and the quantities indivisible by their
very nature: people, animals, and other organisms, moments of time, locations of space, particles, some
commodities, digital images and other man-made data, etc. With the help of the calculus machinery called
(cid:16)limits(cid:17), we invariably choose to transition to the continuous part of calculus, especially when we face
continuous phenomena and the quantities in(cid:28)nitely divisible either by their nature or by assumption: time,
space, mass, temperature, money, some commodities, etc. Calculus produces de(cid:28)nitive results and absolute
accuracy (cid:21) but only for problems amenable to its methods! In the classroom, the problems are simpli(cid:28)ed
untiltheybecomemanageable; otherwise, wecirclebacktothediscretemethodsinsearchofapproximations.
Within a typical calculus course, the student simply never gets to complete the (cid:16)circle(cid:17)! Later on, the
graduate is likely to think of calculus only when he sees formulas and rarely when he sees numerical data.
In this book, every concept of calculus is (cid:28)rst introduced in its discrete, (cid:16)pre-limit(cid:17), incarnation (cid:21) elsewhere
typically hidden inside proofs (cid:21) and then used for modeling and applications well before its continuous
counterpart emerges. The properties of the former are discovered (cid:28)rst and then the matching properties of
the latter are found by making the increment smaller and smaller, at the limit:
discrete continuous
∆x→0
−−−−−−−−−−→
calculus calculus
The volume and chapter references for Calculus Illustrated 7
Calculus Illustrated
The volume and chapter references for
This book is a part of the series Calculus Illustrated. The series covers the standard material of the under-
graduate calculus with a substantial review of precalculus and a preview of elementary ordinary and partial
di(cid:27)erential equations. Below is the list of the books of the series, their chapters, and the way the present
book (parenthetically) references them.
(cid:4) Calculus Illustrated. Volume 1: Precalculus
1 PC-1 Calculus of sequences
1 PC-2 Sets and functions
1 PC-3 Compositions of functions
1 PC-4 Classes of functions
1 PC-5 Algebra and geometry
(cid:4) Calculus Illustrated. Volume 2: Di(cid:27)erential Calculus
2 DC-1 Limits of sequences
2 DC-2 Limits and continuity
2 DC-3 The derivative
2 DC-4 Di(cid:27)erentiation
2 DC-5 The main theorems of di(cid:27)erential calculus
2 DC-6 What we can do with calculus
(cid:4) Calculus Illustrated. Volume 3: Integral Calculus
3 IC-1 The Riemann integral
3 IC-2 Integration
3 IC-3 What we can so with integral calculus
3 IC-4 Several variables
3 IC-5 Series
(cid:4) Calculus Illustrated. Volume 4: Calculus in Higher Dimensions
4 HD-1 Functions in multidimensional spaces
4 HD-2 Parametric curves
4 HD-3 Functions of several variables
4 HD-4 The gradient
4 HD-5 The integral
4 HD-6 Vector (cid:28)elds
(cid:4) Calculus Illustrated. Volume 5: Di(cid:27)erential Equations
5 DE-1 Ordinary di(cid:27)erential equations
5 DE-2 Vector variables
5 DE-3 Vector and complex variables
5 DE-4 Systems of ODEs
5 DE-5 Applications of ODEs
5 DE-6 Partial di(cid:27)erential equations
Each volume can be read independently.
The volume and chapter references for Calculus Illustrated 8
A possible sequence of chapters is presented below. An arrow from A to B means that chapter B shouldn’t
be read before chapter A.
About the author 9
About the author
Peter Saveliev is a professor of mathematics at Marshall University, Hunt-
ington, West Virginia, USA. After a Ph.D. from the University of Illinois at
Urbana-Champaign, he devoted the next 20 years to teaching mathematics.
Peter is the author of a graduate textbook Topology Illustrated published
in 2016. He has also been involved in research in algebraic topology and
several other (cid:28)elds. His non-academic projects have been: digital image
analysis, automated (cid:28)ngerprint identi(cid:28)cation, and image matching for mis-
sile navigation/guidance.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
(cid:4) Chapter 1: Functions in multidimensional spaces . . . . . . . . . . . . . . . . . . . . . . . 13
1.1 Multiple variables, multiple dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Euclidean spaces and Cartesian systems of dimensions 1, 2, 3,... . . . . . . . . . . . . . . . . . 18
1.3 Geometry of distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.4 Where vectors come from . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.5 Vectors in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.6 Algebra of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.7 Convex, a(cid:30)ne, and linear combinations of vectors . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.8 The magnitude of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
1.9 Parametric curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
1.10 The angles between vectors; the dot product . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
1.11 Projections and decompositions of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
1.12 Sequences and topology in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
1.13 The coordinatewise treatment of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
1.14 Partitions of the Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
1.15 Discrete forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
(cid:4) Chapter 2: Parametric curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.1 Parametric curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
2.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
2.4 Location - velocity - acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
2.5 The change and the rate of change: the di(cid:27)erence and the di(cid:27)erence quotient . . . . . . . . . . 158
2.6 The instantaneous rate of change: derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
2.7 Computing derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
2.8 Properties of di(cid:27)erence quotients and derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 169
2.9 Compositions and the Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
2.10 What the derivative says about the di(cid:27)erence quotient: the Mean Value Theorem . . . . . . . 177
2.11 Sums and integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
2.12 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
2.13 Algebraic properties of sums and integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
2.14 The rate of change of the rate of change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
2.15 Reversing di(cid:27)erentiation: antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
2.16 The speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
2.17 The curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
2.18 The arc-length parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
2.19 Re-parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
2.20 Lengths of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
2.21 Arc-length integrals: weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
2.22 The helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
(cid:4) Chapter 3: Functions of several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
3.1 Overview of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
3.2 Linear functions: lines in R2 and planes in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
3.3 An example of a non-linear function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
10