Gottfried Wilhelm Leibniz 16461716

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Gottfried Wilhelm Leibniz (1646-1716)

• Born July 1, 1646, in Leipzig
• 1661, entered University of Leipzig (as a law
  student)
• 1663, baccalaureate thesis, De Principio
  Individui (On the Principle of the Individual')
• 1667, entered the service of the Baron of
  Boineburg
• 1672 - 1676, lived in Paris (met Malebranche,
  Arnauld, Huygens)
• 1675, laid the foundation of the
  differential/integral calculus
• 1676, entered the service of the Duke of
  Hannover worked on hydraulic presses, windmills,
  lamps, submarines, clocks, carriages, water
  pumps, the binary number system
• 1684 published Nova Methodus Pro Maximus et
  Minimus (New Method for the Greatest and the
  Least'), an exposition of his differential
  calculus
• 1685, took on the duties of historian for the
  House of Brunswick
• 1691, named librarian at Wolfenbuettel
• 1700, named foreign member of the French Academy
  of Sciences in Paris
• 1711, met the Russian czar Peter the Great
• Died, November 14, 1716, in Hannover

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Gottfried Wilhelm Leibniz (1646-1716)

• De Arte Combinatoria (On the Art of
  Combination), 1666
• Hypothesis Physica Nova (New Physical
  Hypothesis), 1671
• Discours de métaphysique (Discourse on
  Metphysics), 1686
• unpublished manuscripts on the calculus of
  concepts, c. 1690
• Nouveaux Essais sur L'entendement humaine (New
  Essays on Human Understanding), 1705
• Théodicée (Theodicy), 1710
• Monadologia (The Monadology), 1714

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Gottfried Wilhelm Leibniz (1646-1716)

• 1675, on 21 November he wrote a manuscript using
  the ?f(x) dx notation for the first time. The ?
  stands for an elongated S, for sum (summa). The
  dx stands for an infinitesimal difference. Also
  the product rule for differentiation is given.
• 1676 in autumn Leibniz discovered the familiar
  d(xn) nxn-1dx for both integral and fractional
  n.
• 1684 published Nova Methodus Pro Maximus et
  Minimus (New Method for the Greatest and the
  Least'), an exposition of his differential
  calculus
• In 1686 Leibniz published, in Acta Eruditorum, a
  paper dealing with the integral calculus with the
  first appearance in print of the ? notation and a
  proof of the Fundamental Theorem

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Gottfried Wilhelm Leibniz (1646-1716)

• Three basic inputs for Leibnizs work on integral
  calculus
• In all his studies he was striving for a
  universal language.
• His study of series. Forming differences and
  taking partial sums are inverse operations.
• The idea of a characteristic triangle which has
  infinitesimal sides.
• ad 1. Unlike Newton, Leibniz thought a lot about
  the way to present his ideas in a good formalism.
  In fact, his notation is still used today. Note
  that good notation is key for dealing with
  complex problems.
• ad 3. Leibniz constructed an infinitesimal
  triangle whose curved hypotenuse approximates
  the derivative and used this construction to give
  an integration. In essence he solves the problem
  of integration via the fundamental theorem.

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Gottfried Wilhelm Leibniz (1646-1716)

• ad 2. Consider a series tn for n in N. Also let
  tn be given by
• tn tn1-tn and
• sn Si1,..,nti then
• sntn1-t1.
• Compare this to the fundamental theorem!
• Example triangular numbers i(i1)/2
• tn-2/n
• tn2/n-2/(n1) 2/(n(n1))
• sn2-2/(n1) and as n ? 8 then sn ? 2.
• Compare to
• sn behaves like an integral
• the limit n ? 8 corresponds to the indefinite
  integral and
• tn tn1-tn behaves like a derivative.
• Consider the function f with f(n)tn then
• t(f(n1)-f(n))/1 and with Dx1
• t(f(nDx)-f(n))/Dx
• which is what is called today the discrete
  derivative.

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Gottfried Wilhelm Leibniz (1646-1716)

• We will follow Leibniz to consider the quadratrix
  C(C) of a given curve H(H).
• C(C) is given by its law of tangency. I.e. it is
  the curve which at each point has a prescribed
  tangent. (This statement is essentially the
  Fundamental Theorem).
• Given an axis and an ordinate Leibniz associates
  to each point C on the curve C(C) two triangles
• The assignable triangle CBT composed out of the
  axis, the ordinate and the tangent to the point.
• The in-assignable triangle GLC composed out of an
  infinitesimal piece parallel to the axis, an
  infinitesimal piece parallel the ordinate and an
  infinitesimal piece of the curve. The two other
  sides of the triangle are dx and dy.

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Gottfried Wilhelm Leibniz (1646-1716)

• Given the curve H(H) fix the coordinates AF and
  AB. Let C(C) be the curve, s.t. C is on HF and
  TBCB HFa, where CB is parallel to AF and CT
  is the tangent to C(C) at C and a is a constant
  setting the scale. Or since BTAF
• axBTAFxFHA(rectangle AFH)
• Theorem. Let (F)(C) be a parallel to FC. Let E be
  the intersection with CB, (C) be the intersection
  with the curve C(C) and (H) the intersection with
  the curve H(H). Then
• a E(C)A(region F(H))
• i.e. aE(C) is the area under the curve H(H)
  above F(F) and hence integral from F to (F) of
  H(H).
• Also if A is the intersection of H(H) with AF
  then
• aFCA(region(AFH))
• i.e. aFC is the area under the curve H(H) above
  AF.
• In other words If C(C) satisfies the tangency
  condition it is the quadratrix.
• Proof. Let AFy, FHz, BTt and FCx, then
• tzya (see above)
• ty dxdy (yAFBC and dxdyBTBC)
• So a dxz dy and ?a dx ?z dy thus ax ?z
  dyAFHA.

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Gottfried Wilhelm Leibniz (1646-1716)

• With the fundamental theorem integration boils
  down to finding an anti-derivative.
• Leibniz had a calculus for dealing with
  derivatives in terms of infinitesimals. This is
  still used in physics.
• Example xy3/3 and xdx(ydy)3/3 so
• With the fundamental theorem we thus again
  squared the parabola.
• The use of the infinitesimals dx is however
  tricky. One has to claim that in the last line dy
  and (dy)2 are zero, without making dx or dy zero
  in any previous line. This can be rigorously
  achieved by non-standard analysis, but this had
  to wait 300 years. The next step historically was
  to introduce limits.

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