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 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

    • 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}

      $$

      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 IX

    See Chain rule

    • Exercises VI
    • Exercises VII

  • 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.
    • 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