Calculus

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© 1998-2008 by Glenn Elert -- A Work in Progress
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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?

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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 + Δx). The smaller the distance between the points, the better the approximation. The limit of this procedure as Δx 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(xi) Δx
		Δx → 0 n → ∞
			i = 1

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

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The area under a curve y = ƒ(x) can be approximated by adding rectangles of width Δx and height ƒ(x). The more rectangles (or equivalently, the narrower the rectangles) the better the approximation. The limit of this procedure as Δx 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 Wilhelm von Leibniz (1646-1716) Germany of Sorbian (Slavic) descent
fluxions, a term that is not much used anymore	∫ (elongated s) from Latin summa, sum d 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	short s, final s	long s, medial s	integral symbol

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
• 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?

Summary

• bullet

Problems

practice

1. Write something.
   • Answer it.
2. Write something.
   • Answer it.
3. Write something.
   • Answer it.
4. Write something completely different.
   • Answer it.

numerical

1. problems

Resources

• general
  • The Integrator, Wolfram Research
  • The Mechanical Universe and Beyond (video on demand, login required)
    • Derivatives, The function of mathematics in physical science and the derivative as a practical tool.
    • Integration, Newton and Leibniz arrive at the conclusion that differentiation and integration are inverse processes.

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