Silvanus Thompson, (Thompson 1914)

## Summary¶

An introductory book on Calculus which focuses on easing the reader into the material and establishing their intuition.

## Thoughts¶

### Skeleton¶

#### Front Matter¶

• Boilerplate
• Transcriber’s Note

What one fool can do, another can.

• Preface
• Prologue

#### Main Matter¶

• 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 nth-degree 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 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

\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}

Product rule

\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}

• 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 Die-away 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.
• Exercises XVIII
• Chapter XX

• Exercises XIX
• Chapter XXI

#### Back Matter¶

• Epilogue and Apologue
• Table of Standard Forms