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Newton
Newton was born Christmas day 1642
Started in Cambridge 1661, BA early in 1665
Successor of Barrow 1669
Warden of Mint in 1696
Died and buried in 1727, with pump just as a king
Started Euclid's elements and Descartes' geoemtry in 1664
worked in isolation 1665/1666 because of plague
laid foundation of calculus, nature of light, and theory of
gravitation.
Work on mathematics largely unpublished.
In 1666 Newton used fluxions, tract on fluxions, his way to
work with infinitesimals.
Fluxions
Newton used fluxions very successfully in understanding
and working with tangent vector. The modern and
important way of understanding fluxions is to assume a
time parametrization of very variable used. Moreover,
almost Einstein like one can make a change of variable
using other parameters than time.
As an illustration we assume that a particle moves on a
curve (x(t),f(x(t))=(x(t),y(t)). Then the slope of the tangent
line is given by
Geometric
interpretation
similar as done
by Robertval!
The same method applies to implicitly
defined functions. Assume that
f(x(t),y(t))=0
Then
The fundamental theorem of calculus
The fundamental theorem was discovered
by Newton as an inverse problem:
Find y given a relation between x and
Newton's calculus
Newton used a new infinitesimal tool so
called fluctions and invented large
parts of modern calculus (Calculus I/II)
He formulated the fundamental
theorem
of calculus, including change of
variables and integration by parts.
He discovered the series expansion for
This was even new in this compact
form for integers alpha, let alone
arbitrary real numbers.
Newton was an expert on series. He
combined series with integration
techniques and frequently interchanged
summation and integration.
The starting point for some of his
Anlaysis is Mercators series development
for the natural logarithm and Wallis'
interpolation scheme.
This was combined with his method to
find the inverse of a power series with
the help of term by term calculation.
Indeed, for a power series of the form
Now, he has a power series and can
compute a number of terms. Then he
applies his inversion method and finds
He does that up to five or seven terms
and finds the factorials. That can
certainly not be a coincidence!
Quadratix:
The intersection point satisfies
Integration by parts
Let us assume that two functions
v=v(x) and y=y(z)
are given.
The fundamental theorem of calculus
says that for the area functions
we have
Now we assume that there is an internal relations ship
between x and z such that
holds. Then we get:
This means
This argument show that if the integrals are the same
then
y(x(z))= v(z) dx/dz(z)
The converse is also true: if the relation between x
and z is such that
Then
Example:
Newton then generalizes to the situation where
t=F(x,v)+ ks
and find