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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 Work on mathematics largely unpublished.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