Text Version
Copyright © 2004, by Bluejay Lispware, all rights
reserved.
This material may be reproduced for personal use,
but may only be distributed within an institution if the
institution holds a current Mathwright Library Institutional License
The following 500 page
book, which consists of 49 lectures and 43 Interactive Explorations, is presented
in 10 Mathwright Microworlds. All of the text is contained within the Microworlds,
just as it is on this page.
This book is designed
to illuminate, and to give readers the chance to explore in some depth, the
basic ideas of Calculus within the context of its first triumphal scientific
success: Isaac Newton's deduction of Kepler's three laws of planetary motion
from his single hypothesis of Universal Gravitation. This book differs
from a Calculus textbook in several ways. While it assumes a basic understanding
of Geometry and Algebra, it is designed around the theme of Gravitation,
rather than any particular syllabus. It is an Interactive Story that
invites you to explore a selected range of ideas from the Calculus that were
inspired by this theme.
Unlike a text, the book
does not attempt to give an encyclopedic account of all the standard techniques
of calculation and problem-solving that readers might someday be called upon
to know. We develop in some detail a great variety of techniques, but only as
we require them in the telling of the story. To that end, each Microworld
Section of each Chapter of the book first discusses a problem
that we need to solve to deepen our understanding of the gravitation theme,
and then recruits and explains the techniques that Calculus can supply
to help us solve it. The problems are not easy ones, but we attempt in the lectures
and interactive explorations to bring them to life, so that readers can experiment,
and become familiar with them.
The book is written
for readers who enjoy mathematics, and have the curiosity and
the desire to see the small part of it that we develop here, as a whole:
roots, branches, and leaves. As the title implies, this book is not meant to
be read like a text, but is designed for you to learn by acting
on the various stages of the story. If you would like to know what Calculus
is really about, and how it came to be, this book may be for you.
But
we begin with a word of caution. This book is a gradual ascent to a high place.
We have tried to write it so that the early part will be immediately meaningful
to the thoughtful and curious high-school student. The Pre-Calculus Introduction
is fairly non-technical, dealing mainly with the circle of ideas that led to
the invention of Calculus in the 17th century. Those ideas are what really matter,
and will form the basis for all that follows in later chapters. And as we gather
the analytic tools and techniques that Isaac Newton invented to solve Kepler's
problem of planetary motion, we will adopt an increasingly "rigorous"
tone. The level of discourse may then not be accessible to beginners, no matter
how dedicated they are to the task. It is an unfortunate myth that a beginning
student can master the calculus after only one or two years of study.
When you have followed the story that this book tells, you will be well on your
way to that mastery!
So
we counsel patience. This book is a Story, not an encyclopedia. It is not
a textbook. Take your time, and enjoy it. You will not find the answers
neatly laid out in its pages, until you yourself ask the questions.
And that requires both time and reflection. There is no "royal road"
to the Calculus, and you may find that you return to this book many times in
order to see a point clearly. But we believe that the book will offer you a
new opportunity to formulate and to ask your questions. Each part of each section
of each chapter invites you to experiment, and to bring the ideas to life in
a way that is meaningful to you.
We
introduce the key ideas of the Calculus only as we need them to solve the problem
at hand, whether it is to calculate the trajectory of a baseball, to determine
the escape velocity, or to place a satellite into geosynchronous orbit around
the Earth. And like the Calculus itself, our problems all grow out of the questions:
"How does an object fall ? How do the Moon and Planets move ?" That
is the theme of the story, our main question, for which the Calculus is the
language that provides our best answer.
In the hands of its
creator, Isaac Newton, the Calculus was a musical instrument. He made it sing
the song that told of Kepler's "harmony of the spheres," and he gave
us an instrument of thought that he felt would reveal the deepest secrets of
Nature. In this series of lectures, you will follow young Isaac, and will see
for yourself what music Calculus can make in your own hands.
The microworld explorations
that constitute the Chapter Sections will also invite you to roll up your sleeves
and to use the Calculus to solve problems. Each lecture of a microworld Section
has at least one separate exploration. Scroll down to the end of each lecture
to enter its microworld. In
fact, all of these lectures will be found in the microworlds themselves. Each
Part of each Microworld, including the Calculator page has the story for that
page under the 
icon.
Just click on this icon to read the story for the page. You may print the lecture
page from within the microworld or from this text version. The number of print
pages for each lecture is also indicated below. Finally, the index will take
you quickly to a topic of interest.
Here, you may follow
the hyperlinks to read the text online in your browser. Library
Members may download the PDF file 
for each lecture. Select Save from the Adobe Reader menu, and save the
file to a convenient location on your disk so that in the future you can read
it without having to go online. If you download the Mathwright32 Reader Version
of the Microworld from its Title Page, you will also be able to read and interact
in the Microworld offline. There will be no need for the PDF file in that case,
because all of the lectures will be contained in the Microworld, and, believe
it or not, the Microworlds are much smaller that the PDF files.
Each lecture may be
printed from either place. The Explorations are designed to encourage readers
to pursue their own ideas by asking "What if?" questions. They also
play a heuristic role to help the reader visualize new constructions, techniques
and concepts, but for the reader for whom this book is written, that role is
ancillary. This book is written for active (and aggressive) learners, who by
now should have little trouble visualizing the constructions at the base of
the Calculus. For those readers, the examples will be the important thing, and
we bring the computer environment into a new relationship to the mathematics
for them. In fact, this will likely strike students as a new kind of mathematics
book, unlike one they have ever seen, and certainly not like any Calculus Textbook,
either in content or in form. Teachers: Be prepared for a few
surprises! Library members are invited to correspond with the
author on any questions about the material
or the interactive explorations.
The author gratefully
acknowledges the encouragement, advice, and sometimes the code given by several
colleagues, Mathwright Authors all. Their contributions were both to the spirit
and to the form of this book. In alphabetical order, these are: Margie Hale,
Dan Kalman, Ravinder Kumar, Samad Mortabit, and Mike Pepe. Thank you all.
Index
of Topics and Explorations:
| A |
J |
| Acceleration,
angular |
|
| Acceleration,
centripetal for uniform motion |
K |
| Acceleration,
radial |
Kepler,
John |
| Acceleration
of a polar curve in moving frame coordinates |
Kepler's
Laws |
| Acceleration,
uniform |
Kepler's
Laws, dynamic aspect |
| Acceleration,
uniform, inclined plane |
Kepler's
First Law, deduction |
| Algebra:
Calculator Instructions |
Kepler's
Second Law, deduction |
| Alternating
2-linear map |
Kepler's
Third Law, deduction |
| Alternating
3-linear map |
Kepler's
Second Law: EXPLORATION |
| Anti-differentiation
(Galileo example) |
Kepler'
Third Law, EXPLORATION |
| Antiderivatives
and integrals |
Kinetic
Energy |
| Angle
between vectors |
Kinetic
Energy (inclined planes) |
| Angular
momentum interpreted via cross product |
|
| Approximating
function, general (for integration) |
L |
| |
Limit
Definitions |
| |
Limit
Rules |
| Approximating
function, simple (for integration) |
Limits
of Sequences: EXPLORATION |
| Arc
length and Integration: EXPLORATION |
Linear
coordinates |
| Arc
length and rectifiability |
Linear
transformation |
| Arc
length function definition |
Linear
transformation of the plane, determinant |
| Arc
length parameterization |
Linear
transformation of the plane, matrix |
| Area
and volume integrals |
Log-log
plots |
| Areas
and Logarithms: EXPLORATION |
Logarithms |
| Arithmetic:
Calculator Instructions |
Logarithms,
natural |
| Average
rate of change |
Logistic
map |
| |
|
| B |
Logistic
map and bifurcation: EXPLORATION |
| Ballistic
Projectile, Maximizing the range |
|
| Bernoulli's
Problem |
M |
| Bernoulli
Slider: EXPLORATION |
Manifolds |
| Boltzmann
distribution |
Mars,
retrograde motion |
| Brachistchrone |
Matrix:
Calculator Instructions |
| |
Matrix
algebra |
| C |
Matrix
Determinant (2x2 matrix) |
| Calculator
Instructions |
Matrix
of a linear transformation |
| |
Maxima,
Minima and Inflections: EXPLORATION |
| Calculus:
Calculator Instructions |
Maximizing
the range with Newton's Method: EXPLORATION |
| Cartesian
and Polar coordinates: EXPLORATION |
Mean
Value Theorem |
| Cartesian
coordinates |
Measurement
strategy |
| Cauchy
convergence criterion |
Metric
properties of curves |
| Cauchy-Schwartz
Inequality |
Metric
structure in the plane |
| Center
of Mass |
Momentum
as a Vector: EXPLORATION |
| Chambered
Nautilus, gnomons, and fractals |
Momentum
conservation |
| Chaos
and the Logistic map |
Momentum
Principle (Galileo and Newton) |
| |
Monotonicity |
| Change
of integration variable |
Moving
Frame of Polar curves |
| Change
of integration variable formula |
Moving
Frame solutions of second order equations |
| Change
of linear coordinates |
|
| Change
of variable strategy |
N |
| Complex
numbers as matrix algebra |
Natural
logarithms as area integrals |
| Compound
Interest |
Newton,
Isaac (on conservation of angular momentum) |
| Concavity
|
|
| Conic
Section (description) |
Newton,
Isaac (on conservation of energy) |
| Conservation
Laws, combining |
Newton,
Isaac (on Kepler's First Law) |
| Conservation
of Angular Momentum |
Newton,
Isaac (on Kepler's Second Law) |
| Conservation
of Angular Momentum deduction |
Newton,
Isaac (on Kepler's Third Law) |
| Conservation
of Angular Momentum via cross product |
Newton's
Gravitational Law (simple form) |
| Conservation
of Energy (Dimension 1) |
Newton's
Universal Gravitational Law (vector form) |
| Conservation
of Energy (Dimension 2 -- Circularity conditions) |
Newton's
Method: EXPLORATION |
| Conservation
of Energy (Dimension 2 -- Reciprocity principle) |
Newton's
Method and Chaos: EXPLORATION |
| Conservation
of Energy (inclined planes) |
Newton's
Recursive Method |
| Conservation
of Energy (pendulum) |
|
| Conservation
of Kinetic Energy |
O |
| Conservation
of Linear Momentum |
Orientation
preserving isometries in the plane |
| Continuity
|
|
| Coordinate
transformation, active |
Orthogonal
projection of one vector on another |
| Coordinate
transformation, passive |
Orthonormal
basis |
| Critical
Points |
|
| Cross
Product of vectors in 3 dimensions |
Orthonormalization
in 2 dimensions |
| Curvature
as angular acceleration (arc length parameterization) |
Orthonormalization
in 3 dimensions |
| Curvature
integral of a Plane curve |
Orientation
in space |
| Curvature
of a plane curve |
Osculating
Circle |
| Curvature
of a plane curve as metric property |
|
| Curve
bending, geometric measure of |
P |
| Curve,
polygonal approximation |
Parabola |
| Curve,
rectifiability |
Parallelogram
law for vector addition |
| Curve
turning, geometric measure of |
Parallelolopiped
volume |
| Curve
parameter changing: EXPLORATION |
Partition
of an interval |
| |
Pendulum
Interrupted: EXPLORATION |
| D |
Pendulum,
nonlinear with frictional damping |
| Derivative
Approximation |
Pendulum
Over the Top: EXPLORATION |
| Derivative
as a Limit |
Perpendicular
projection of one vector on another |
| Derivative,
defined |
Perpendicularity
of vectors |
| Determinant
as signed Parallolopiped volume |
Plane
Curvature and the Osculating Circle: EXPLORATION |
| Determinant
of a 2x2 matrix |
Plane
curve, Integral of the curvature |
| Determinant
of a 3x3 matrix |
Plane
planetary motion deduced |
| Determinant
of a linear transformation of space |
Planetarium
Visit: EXPLORATION |
| Determinant
of a linear transformation of the plane |
Polar
coordinates |
| Difference
Quotient |
Polar
functions and curves |
| Differential
Approximation |
Polygonal
approximation of a curve |
| Differential
Approximation: Maximizing ballistic range |
Position
and Velocity Portrait EXPLORATION |
| Differential
equation strategy |
Position
and Velocity Portraits (Escape Velocity) |
| Differential
equation defined, first order |
Position
Portrait |
| Differential
equation, defined, second order in the line |
Potential
Energy (inclined planes) |
| Differential
equation, first order in the plane |
Ptolemaic
Theory |
| Differential
equation, second order: EXPLORATION |
|
| Differential
equation, linear second order in the line, solved |
|
| Differentiation
Rules |
|
| Discrete
Exponential function and compound interest: EXPLORATION |
|
| Discrete
geometric model of equiangular spiral: EXPLORATION |
|
| Dot
product of vectors in the plane |
|
| Dot
product of vectors in space |
Q |
| |
|
| E |
R |
| e,
powers of: EXPLORATION |
Radius
of Curvature |
| Eccentricity
and Conics: EXPLORATION |
Reciprocity
Principle |
| Eccentricity
and Energy: EXPLORATION |
Riemann
Integration and arc length |
| Eigenvalue,
repeated (second order equation) |
Riemann
Sums |
| Ellipse |
Riemann
Sums, limits of |
| Ellipse,
area |
Rocket
Science 101, Docking with a Satellite: EXPLORATION |
| Ellipse,
Focus-locus property in focus-directrix form |
Rolle's
Theorem |
| Ellipse,
Focus-locus property in Cartesian form |
Rotation
matrix |
| Ellipse,
geometry of |
|
| Energy
conditions for Keplerian orbits |
S |
| Equiangular
spiral and self similarity |
Satellite
orbits, circular |
| Equipotential
curves |
Satellite
Orbits: EXPLORATION |
| Escape
Velocity |
Second
order differential equations |
| Escape
Velocity: EXPLORATION |
Second
order differential equation in the line |
| Escape
Velocity and Conservation of Energy |
Second
order differential equation in the line: EXPLORATION |
| Euclidean
matrix |
Second
order equation, main |
| Euclidean
structure in the plane |
Second
order differential equation in the plane: EXPLORATION |
| Euclidean
structure, covariance under rotations |
Second
order linear differential equations in the line, solved |
| Euler's
Method |
Semi-geometric
series |
| Euler's
Number and Natural Exponentials: EXPLORATION |
Sequences,
limits |
| Euler
Sequence: EXPLORATION |
Sequences,
recursive |
| Exponential
co-variation |
Series,
semi-geometric |
| Exponential
decay |
Simple
Approximating function (for integration) |
| Exponential
function, defined |
Simple
harmonic oscillator equation |
| Exponential
function, discrete |
Simple
harmonic oscillator solution |
| Exponential
functions, graphing: EXPLORATION |
Space
Flight Instructions |
| Exponential
Function, natural |
Sphere,
volume |
| Exponential
Function, standard form |
Sprite
Animation: Calculator Instructions |
| |
|
| F |
T |
| Fibonacci
Sequence |
Total winding and Gauss' Theorem |
| Fibonacci
Sequence and Golden Ratio: EXPLORATION |
Triangle
Inequality |
| Fibonacci
Sequence: recursive definition |
|
| First
order differential equation in the plane |
U |
| Focus-locus
property of ellipse in focus-directrix form |
Uniform
circular motion |
| Focus-locus
property of ellipse in Cartesian form |
Uniform
motion solution (second order equation) |
| Force
Vectors |
Uniform
change |
| Freely
Falling Objects |
Universal
Law of Gravitation |
| Frenet
Frame of a space curve |
|
| Frenet
Frames of space curves: EXPLORATION |
V |
| Fundamental
Theorem of Calculus |
Vector
addition |
| |
Vector
addition, covariance of |
| G |
Vector,
angle between |
| Galileo
Galilei |
Vector
Cross Product in 3 dimensions |
| Galileo's
Experiment: EXPLORATION |
Vector
Dot Product in 3 dimensions |
| Galileo's
Formula |
Vector
Dot Product in 2 dimensions |
| Gauss
Map: EXPLORATION |
Vector
length |
| Gauss
Map for a plane curve |
Vector
perpendicularity |
| Gauss'
Theorem and total winding |
Vector
projection of one vector on another |
| General
Approximating function (for integration) |
Vector
space (2 dimensional) |
| Geometric
Invariants of plane vectors: EXPLORATION |
Vectors
|
| Geometric
object, curve as |
VelocityCurve |
| Geometric
and semi-geometric series: EXPLORATION |
Velocity
Moving Frames |
| Geosynchronous
Satellite Orbit |
Velocity
Portrait |
| Golden
Ratio |
Vertical
Projectiles: EXPLORATION |
| Golden
Rectangle |
|
| Gram-Schmidt
orthonormalization in 2 dimensions |
W |
| Gram-Schmidt
orthonormalization in 3 dimensions |
Winding
number |
| Graphing:
Calculator Instructions |
Winding
number and Gauss' Theorem |
| Graphing,
Linear Best Fit |
|
| Graphing,
Log-log plots |
X |
| Graphing
Polar Curves: EXPLORATION |
|
| Gravitational
Potential Wells: EXPLORATION |
Y |
| Gravity,
Newton's Universal Law of Gravitation |
|
| |
Z |
| H |
|
| Hyperbola |
|
| |
|
| I |
|
| Inclined
Plane |
|
| Inclined
Plane: A Curious Constant: EXPLORATION |
|
| Inclined
Plane:Conservation of Energy |
|
| Induction,
Method of |
|
| Inertial
Coordinate Frame |
|
| Inflection
Point |
|
| Inner
Product structure in the plane |
|
| Inner
Product structure in 3 dimensions |
|
| Inner
Product structure, covariance under rotations |
|
| Integral
as limit of Riemann Sums |
|
| Integral
curvature and manifolds |
|
| Integral
of the curvature of a plane curve |
|
| Integration
as method of approximation |
|
| Integration,
change of variable formula |
|
| Integration,
change of variable strategy |
|
| Integration
strategy |
|
| Irrational
numbers in measurement |
|
| Isometry
of the plane |
|
| |
|
