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  f(x) = 

f(x + Δx) − f(x)  
dx  Δx 
keywords: derivative, differentiation, anything else?
The slope of the line tangent to a curve y = f(x) can be approximated by the slope of a line connecting f(x) to f(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)
⌠ ⌡ 
f(x) dx = 


f(x_{i}) Δx 
keywords: integral, integration, indefinite integral, definite integral, limits of integration, more?
The area under a curve y = f(x) can be approximated by adding rectangles of width Δx and height f(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.
f(x) =  ⌠ ⌡ 
⎛ ⎝ 
d  f(x)  ⎞ ⎠ 
dx =  d  ⎛ ⎝ 
⌠ ⌡ 
f(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, 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  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?