Transcript The Fundamental Theorem of Calculus: History, Intuition

The Fundamental Theorem of Calculus:
History, Intuition, Pedagogy, Proof.
V. Frederick Rickey
West Point
SUNY/Oneonta, October 8, 2010
Isaac Newton
1642 - 1727
• 1702 portrait by
Kneller
• The original is in the
National Portrait
Gallery in London
Newton’s Mathematical Readings
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Barrow
Oughtred
Descartes
Schooten
Viete
Wallis
Wallis
Euclid (1655)
Clavis (1652)
2nd Latin (1659-60)
Exercitationum (1657)
Opera (1646)
Arithmetica infinitorum (1655)
Tractatus duo (1659)
Took Descartes’s Geometry in hand, tho
he had been told it would be very
difficult, read some ten pages in it, then
stopt, began again, went a little farther
than the first time, stopt again, went
back again to the beginning, read on til
by degrees he made himself master of
the whole, to that degree that he
understood Descartes’s Geometry better
than he had done Euclid.
Descartes’s Geometry, 1637, 1659
Descartes adopted Aristotle’s dictum
The proportion between straight lines
and curves is not known and I even
believe that it can never be known by
man.
van Heuraet on Arc Length, 1659
van Heuraet’s rectification, 1659
Rectification Destroyed
• Aristotle’s dictum
and
• Descartes’ program
• But the story ends well.
The Fundamental Theorem of Calculus
A Method whereby
to square such
crooked lines as
may be squared.
• van Heuraet swapped arc length for area
• Newton swapped area for a tangent
For Newton
•
Mathematical quantities are described by
Continuous Motion.
– E.g., Curves are generated by moving points
•
In Modern Terms: All variables are functions
of time
• Newton said that quantities flow, and so
called them fluents.
• How fast they flow – or flex – he called
fluxions.
• Par abuse de langu,
d/dt ( fluent ) = fluxion
Given an equation
involving any number of
fluent quantities to find
the fluxions and vice
versa.
Gottfried Wilhelm
von Leibniz
(1646 – 1716)
The Nova methodus
of 1684 – the first
paper on the
differential calculus.
Leibniz proves
first FTC in 1690
The Fundamental Theorem
of Calculus
Newton:
Leibniz:
The Windshield
Wiper Model
The Sideways
Chalk Model
The Isochrone Problem
• Find a curve along which a body will descend
equal distances in equal times
• Johann Bernoulli reduces it to the Differential
Equation √a dx = √y dy.
• Et eorum integralia !
• The curve is a semi-cubical parabola,
y3 = 9/4 a x2
Johann Bernoulli in 1743
His spirit sees truth
His heart knows justice
He is an honor to the Swiss
And to all of humanity
• Voltaire
Finding areas under curves
Decompose the region into
infinitely many differential
areas
1. with parallel lines
2. with lines emanating from
a point
3. with tangent lines
4. with normal lines.
We seek the curve where the square of the ordinate BC is the mean
proportional between the square of the given length E and the
curvilinear figure ABC.
E2 / BC2 = BC2 / Area ABC
Area ABC = y4 / a2
By FTC,
y dx = 4 y3 dy / a2
Divide by y and integrate
To get a cubical parabola
Johann Bernoulli’s
definition of an integral
We have previously shown how to
find the differential of a given
quantity. Now we show inversely how
to find the integral of a differential,
i.e., find the quantity from which the
differential originates.
E2 / BC2 = BC2 / Area ABC
Area ABC = y4 / a2
By FTC,
y dx = 4 y3 dy / a2
Divide by y and integrate
To get a cubical parabola
I misread this text.
Bernoulli does NOT use FTC
but only the notion that
an integral is an antiderivative.
• Johann
Bernoulli’s best
student !
• Leonhard Euler
• 1707-1783
Euler about 1737, age 30
• Painting by J. Brucker
• 1737 mezzotint by
Sokolov
• Black below and above
right eye
• Fluid around eye is
infected
• “Eye will shrink and
become a raisin”
• Ask your
ophthalmologist
•
Thanks to Florence Fasanelli
Euler’s Calculus Books
• 1748
Introductio in analysin infinitorum
399
402
• 1755
Institutiones calculi differentialis
676
• 1768
Institutiones calculi integralis
462
542
508
_____
2982
• Defines the integral
as an antiderivative.
• Gives a careful
discussion of
approximating a
definite integral with
a sum of rectangles.
Read Euler,
read Euler,
he is our teacher in everything.
Lisez Euler,
lisez Euler,
c'est notre maître à tous.
Laplace as quoted by Libri, 1846
th
18
In the
century
there was no FTC.
Augustin Cauchy
This famous work of 1821
began to introduce rigor
into the calculus by defining
limits, continuity and
derivatives and proving
theorems about them.
It was never used as a text.
1789 - 1857
Augustin Cauchy
1823
In his Résumé of 1823, Cauchy
• Gave a careful definition as the limit of a sum
of areas of rectangles (evaluated at left
endpoints).
• Proved that the integral of a continuous
function exists.
• Proved the First FTC in a rigorous way.
• Cauchy’s definition of the integral is a
radical break with the past!
• Euler used left sums.
• Lacroix and Poisson tried to prove
the sums converge.
• Fourier needed to think of the
definite integral as an area.
At the end of the 19th century, authors had two
choices regarding the introduction of the integral:
Either one might define the integral as the limit of a
certain sum, or, alternatively,
integration is the inverse operation to differentiation.
If the former introduction is chosen, then one must
justify some form of the Fundamental Theorem of
Calculus;
if the latter, then the use of the integral in applications
becomes the sticking point.
Rosenstein and Temelli, 2001
The Name “FTC”
in Research Monographs.
• Eduard Goursat uses the term “Fundamental
Theorem of Calculus“ in his Cours d'analyse
mathematiques (1902).
• Ernest W. Hobson in his Theory of Functions of a Real
Variable (1907) has a chapter entitled “The
fundamental theorem of the integral calculus.”
• Vallee Poussin in his Cours d'analyse innitesimale
(1921) uses the name “relation fondamentale pour le
calcul des integrals "
William Anthony Granville
• Text used at West Point
1907-1948 and
1953-1963
Granville 1911
• Granville and Smith define indefinite
integration as antidifferentation.
• The definite integral is defined as F(b) – F(a),
where F’(x) = f(x).
• Thus there is no FTC in our modern sense.
• They use FTC in the sense of du BoisReymond, 1876, 1880.
The Name “FTC” in Textbooks
• Björling (1877) gives the name “Grundsats” to
the second fundamental theorem in a Swedish
textbook
• G. H. Hardy, A Course of Pure Mathematics
(1908), uses the phrase and provides a proof.
• George Thomas uses the phrase in his Calculus
(1951).
First use of the name
2nd FTC, 1958
To be continued . . .
after much more research.