# Calculus

## Discussion

### the derivative

only straight lines have the characteristic known as slope

instantaneous rate of change, that is, the slope of a line tangent to the curve

d |
ƒ(x) = |
lim | ƒ(x + Δx) − ƒ(x) |

dx |
Δx → 0 |
Δx |

keywords: derivative, differentiation, anything else?

The slope of the line tangent to a curve *y* = ƒ(*x*) can be approximated by the slope of a line connecting ƒ(*x*) to ƒ(*x* + Δ*x*). The smaller the distance between the points, the better the approximation. The limit of this procedure as Δ*x* approaches zero is called the derivative of the function.

### the integral

area under the curve (area between curve and horizontal axis)

_{n} |
||||

⌠ ⌡ |
ƒ(x) dx = |
lim | ∑ | ƒ(x_{i}) Δx |

^{Δx → 0n → ∞} |
||||

^{i = 1} |

keywords: integral, integration, indefinite integral, definite integral, limits of integration, more?

The area under a curve *y* = ƒ(*x*) can be approximated by adding rectangles of width Δ*x* and height ƒ(*x*). The more rectangles (or equivalently, the narrower the rectangles) the better the approximation. The limit of this procedure as Δ*x* approaches zero is called the integral of the function.

### the fundamental theorem of calculus

Differentiation and integration are opposite procedures. The anti derivative is the integral. Proof of this is best left to the experts.

ƒ(x) = |
⌠ ⌡ |
⎛ ⎝ |
d |
ƒ(x) |
⎞ ⎠ |
dx = |
d |
⎛ ⎝ |
⌠ ⌡ |
ƒ(x) dx |
⎞ ⎠ |

dx |
dx |

The necessity of adding a constant when integrating (anti differentiating).

### etc.

Calculus was invented simultaneously and independently…

newton | leibniz |
---|---|

Isaac Newton (1642–1727) England |
Gottfried Leibniz (1646–1716) Germany of Sorbian (Slavic) descent |

fluxions, a term that is not much used anymore |
∫ (elongated s) from Latin summa, sumd from Latin differentia, difference |

"Method of Fluxions" published in 1969 (no joke) |
Was it ever published by Leibniz? |

The word *calculus* (Latin: pebble) becomes calculus (method of calculation) becomes "The Calculus" and then just calculus again.

- Latin: a pebble or stone (used for calculation) Calculus also refers to hard deposits on teeth and mineral concretions like kidney or gall stones. It's also related to the words calcium and chalk.
*Calculus*is the diminutive form of*calx*(chalk, limestone). - A method of computation; any process of reasoning by the use of symbols; any branch of mathematics that may involve calculation. Webster 1913

Symbology

Life, Liberty and the pursuit of Happineſs

f | ƒ | ∫ | ſ | s |

short f | long f | integral symbol |
long s (medial s) |
short s (final s) |

Why these alternate versions of s and f are necessary is a matter of protracted discussion.

### calculus of multiple variables

partial derivatives are good for…

- fields
- scalar fields
- gradient

- vector fields
- divergence
- curl

- scalar fields
- error (propagation of error?)

fancy integrals

- ordinary integral
- area under a curve

- double integral
- volume under a surface

- triple integral
- wowie zowie

more fancy integrals

- line integral
- almost the same as a closed line integral — contour integral

- surface integral
- almost the same as a closed surface integral — say something

- volume integral — think vegetables
- spherical shells — like an onion
- cylindrical shells — like a leek
- disks and washers — like… like… um… here's where I lost the vegetable analogy … like a vegetable sliced into chips?