Silvanus Thompson, (Thompson 1914)
Summary
An introductory book on Calculus which focuses on easing the reader into the material and establishing their intuition.
Thoughts
Notes
Skeleton
Front Matter
 Boilerplate

Transcriber’s Note
What one fool can do, another can.
 Preface
 Prologue
Main Matter

Chapter I

Chapter II
Dealing with degrees of smallness. That is, \(\frac{1}{100}\) as the first degree , \({\frac{1}{100}}^2 = \frac{1}{10,000}\) as the second degree, \({\frac{1}{100}}^3 = \frac{1}{1,000,000}\) as the third degree, and so on. The size of the nthdegree may be negligible such that we can safely ignore it.
An ox might worry about a flea of ordinary size—a small creature of the first order of smallness. But he would probably not trouble himself about a flea’s flea; being of the second order of smallness, it would be negligible. Even a gross of fleas' fleas would not be of much account to the ox.

Chapter III

How to read Differentials.
:NOTER_PAGE: (27 . 0.162729)


Chapter IV

Case 1
Let us begin with the simple expression \(y = x^2\) . Now remember that the fundamental notion about the calculus is the idea of growing. Mathematicians call it varying. Now as \(y\) and \(x^2\) are equal to one another, it is clear that if \(x\) grows, \(x^2\) will also grow. And if \(x^2\) grows, then \(y\) will also grow. What we have got to find out is the proportion between the growing of y and the growing of \(x\). In other words our task is to find out the ratio between \(dy\) and \(dx\), or, in brief, to find the value \(\frac{dy}{dx}\).
$$
\begin{align} y &= x^2 \\ y + dy &= (x + dx)^2 \\ y + dy &= x^2 + 2x(dx) + (dx)^2 \\ y + dy &= x^2 + 2x(dx) \\ x^2 + dy &= x^2 + 2x(dx) \\ dy &= 2x(dx) \\ \frac{dy}{dx} &= 2x \end{align}
$$
 4: “What does (dx)2 mean? Remember that dx meant a bit—a little bit—of x. Then (dx)2 will mean a little bit of a little bit of x; that is, as explained above (p. 4), it is a small quantity of the second order of smallness. It may therefore be discarded as quite inconsiderable in comparison with the other terms.”
 5: \(y = x^2\)
Note that I could write the \(y = x^2\) as \(f(x) = x^2\) and use Lagrange’s notation: \(f'(x) = 2x\).
See Power rule

Case 2
$$
\begin{align} y &= x^3 \\ y + dy &= (x + dx)^3 \\ y + dy &= x^3 +3x^2(dx) + 3x(dx)^2 + (dx)^3 \\ y + dy &= x^3 +3x^2(dx) \\ x^3 + dy &= x^3 +3x^2(dx) \\ dy &= 3x^2(dx) \\ \frac{dy}{dx} &= 3x^2 \end{align}
$$

Case of a negative power
$$
\begin{align} y &= x^{2} \\ y + dy &= (x + dx)^{2} \\ y + dy &= x^{2}(1  \frac{2dx}{x} + \frac{2(2+1)}{1 \times 2}{(\frac{dx}{x})}^2  \text{etc.}) \\ y + dy &= x^{2}  2x^{3}(dx) + 3x^{4}{(dx)}^2  4x^{5}{(dx)}^3 + \text{etc.} \\ y + dy &= x^{2}  2x^{3}(dx) \\ dy &= 2x^{3}(dx) \\ \frac{dy}{dx} &= 2x^{3} \end{align}
$$
 3: Expand by the Binomial theorem
 5: Remove small quantities of higher orders of smallness (e.g. \({(dx)}^2\))
 6: Subtract \(y\) from both sides, remembering that \(y = x^{2}\) from 1.

Case of a fractional power
$$
\begin{align} y &= x^{\frac{1}{2}} \\ \dots \\ \frac{dy}{dx} &= \frac{1}{2}x^{{\frac{1}{2}}} \end{align}
$$
 Exercises I


Chapter V

Added constants
$$
\begin{align} y &= x^3 + 5 \\ y + dy &= {(x + dx)}^3 + 5 \\ &= x^3 + 3x^2dx + 3x{(dx)}^2 + (dx)^3 + 5 \\ &= x^3 + 3x^2dx + 5 \\ &= 3x^2dx\\ \frac{dy}{dx} &= 3x^2 \end{align}
$$
Constants disappear during Differentiation.

Multiplied constants
$$
\begin{align} y &= ax^2 \\ \dots \\ \frac{dy}{dx} &= 2ax \end{align}
$$
Constants disappear during Differentiation.
 Exercises II


Chapter VI

Sum
$$
\begin{align} y &= x^2 + c + ax^4 + b \\ &\dots \\ \frac{dy}{dx} &= 2x + 4ax^3 \end{align}
$$
See Sum rule.

Product
By First principles (i.e. replace with \(y + dy\) and \(x + dx\)):
$$
\begin{align} y &= (x^2 + c) \times (ax^4 + b) \\ y &= ax^6 + acx^4 + bx^2 + bc &\dots \\ \frac{dy}{dx} &= 6ax^5 + 4acx^3 + 2bx \end{align}
$$
$$
\begin{align} y &= z \times w \\ &\dots \\ \frac{dy}{dx} &= z\frac{dw}{dx} + w\frac{dz}{dx} \end{align}
$$
Using Product rule:
$$
\begin{align} y &= (x^2 + c) \times (ax^4 + b) \\ \frac{dy}{dx} &= (x^2 + c)\frac{d(ax^4 + b)}{dx} \times (ax^4 + b)\frac{d(x^2 + c)}{dx} \\ &= (x^2 + c)4ax^3 \times (ax^4 + b)2x \\ &= 4ax^5 + 4acx^3 + 2ax^5 + 2bx \\ &= 6ax^5 + 4acx^3 + 2bx \end{align}
$$

Quotient
 Exercises III


Chapter VII
 Exercises IV

Chapter VIII
 Exercises V

Chapter IX
See Chain rule
 Exercises VI
 Exercises VII

Chapter X
 Exercises VIII

Chapter XI
 Exercises IX

Chapter XII
 Exercises X

Chapter XIII
 Partial Fractions.
 Exercises XI
 Differential of an Inverse Function.

Chapter XIV
 Exercises XII
 The Logarithmic Curve.
 The Dieaway Curve.
 Exercises XIII

Chapter XV
 Second Differential Coefficient of Sine or Cosine.
 Exercises XIV

Chapter XVI
 Maxima and Minima of Functions of two Independent Variables.
 Exercises XV

Chapter XVII
 Slopes of Curves, and the Curves themselves.
 Exercises XVI

Chapter XVIII
 Integration of the Sum or Difference of two Functions.
 How to deal with Constant Terms.
 Some other Integrals.
 On Double and Triple Integrals.
 Exercises XVII

Chapter XIX
 Areas in Polar Coordinates.
 Volumes by Integration.
 On Quadratic Means.
 Exercises XVIII

Chapter XX
 Exercises XIX
 Chapter XXI
Back Matter
 Epilogue and Apologue
 Table of Standard Forms
 Answers
 Catalogue
 Transcriber’s Note
 License