Bernoulli Family
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Johann1667-1748
Daniel1700-1782
Jacob1654-1705
Johann II1710-1790
Nicolaus II1695-1720
Jacob II1759-1789
Johann III1744-1807
BERNOULLI FAMILYSOLO HERMELIN
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BERNOULLI FAMILYSOLO
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Niklaus(1623 – 1708)
Nikolaus I(1687 – 1759)
Jacob I(1655 – 1705)
Nikolaus(1662 – 1716)
Johann(1667 – 1748)
Hieronymus(1669 – 1760)
Eduard(1867 – 1927)
Nikolaus II(1695 – 1726)
Daniel(1700 – 1782)
Johann II(1710 – 1790)
Franz(1705 – 1777)
Eva(1903 – 1995)
CarlAlbrecht
(1868 – 1937)
CarlChristoph
(1861 – 1923)
Jacob II(1759 – 1809
AugustLeonhard
(1879 – 1939)
Elisabeth(1873 – 1935)
Hans(1876 – 1959)
JohannJacob
(1831 – 1913)
CarlGustav
(1834 – 1878)
Johann III(1744 – 1807
Daniel II(1751 – 1834
Nikolas III(1754 – 1841
Nikolaus(1793 – 1876)
Hieronymus(1735 – 1786
Johann Jacob(1769 – 1853
Johann Jacob(1802 – 1892
Emanuel(1776 – 1844)
Johannes(1785 – 1869)
Christoph(1782 – 1863)
Leonhard(1786 – 1852)
Leonhard(1791 – 1871)
August Christoph(1839 – 1921)
Karl Johann (1835 – 1906)
Carl Christoph(1809 – 1884)
Eduard(1819 – 1899)
Theodor(1837 – 1909)
http://de.wikipedia.org/wiki/Bernoulli
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Bernoullianum Basel
http://de.wikipedia.org/wiki/Bernoullianum
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http://curvebank.calstatela.edu/lemniscate
The mathematical lineage in Basel is amazing. James taught his brother John. John taught L'Hôpital the famous rule, but John's best known student was another native of Basel, Leonhard Euler. Daniel Bernoulli would earn 10 prestigious awards from the French Académie Royale des Sciences, a record only matched by Euler. His cousin, Nicholas Bernoulli, was the first to pose the famous St.Petersburg paradox. The chair of mathematics at the University was held by a Bernoulli for over a hundred years (1687 - 1790).
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Basel Bernoullianum from bus station
http://de.wikipedia.org/wiki/Bernoullianum
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Nicolaus Bernoulli (1623-1708), Jacob, Nicolaus and Johann’s father, inherited the spice business in Basel that had been set up by his own father, first in Amsterdam and then in Basel. The family, of Belgium origin, were refugees fleeing from persecution by the Spanish rulers of the Netherlands. Philip, the King of Spain, had sent the Duke of Alba to the Netherlands in 1567 with a large army to punish those opposed to Spanish rule, to enforce adherence to Roman Catholicism, and to re-establish Philip's authority. Alba set up the Council of Troubles which was a court that condemned over 12000 people but most, like the Bernoulli family who were of the Protestant faith, fled the country.
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Jacob Bernoulli was compelled to study philosophy and theology by his parents, which he greatly resented, and he graduated from the University of Basel with a master's degree in philosophy in 1671 and a licentiate in theology in 1676. Jacob
1654-1705In 1676, after taking his theology degree, Bernoulli moved to Geneva where he worked as a tutor. He then traveled to France spending two years studying with the followers of Descartes who were led at this time by Malebranche.
Jacob Bernoulli
René Descartes 1596 - 1650
Robert Boyle1627 - 1691
Nicolas Malebranche1638 - 1715
Johann vanWaveren Hudde
1628 - 1724
Robert Hooke1635 - 1703
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Jacob.html
. In 1681 Bernoulli traveled to the Netherlands where he met many mathematicians including Hudde. Continuing his studies with the leading mathematicians and scientists of Europe he went to England where, among others, he met Boyle and Hooke.
At this time he was deeply interested in astronomy and produced a work giving an incorrect theory of comets. As a result of his travels, Bernoulli began a correspondence with many mathematicians which he carried on over many years.
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Jacob1654-1705
Jacob Bernoulli
Jacob Bernoulli returned to Switzerland and taught mechanics at the University in Basel from 1683, giving a series of important lectures on the mechanics of solids and liquids. Since his degree was in theology it would have been natural for him to turn to the Church, but although he was offered an appointment in the Church he turned it down.
Frans van Schooten1615 - 1660
John Wallis1616 - 1703
Isaac Barrow1630 - 1677
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Jacob.html
René Descartes 1596 - 1650
Bernoulli's real love was for mathematics and theoretical physics and it was in these topics that he taught and researched. During this period he studied the leading mathematical works of his time including Descartes' Géométrie and van Schooten’s additional material in the Latin edition. Jacob Bernoulli also studied the work of Wallis “Arithmetica Infinitorum” and Barrow “Lectiones Geometricae” and through these he became interested in infinitesimal geometry. Jacob began publishing in the journal Acta Eruditorum which was established in Leipzig in 1682.
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Jacob1654-1705
Jacob and Johann Bernoullihttp://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Jacob.html
One of the most significant events concerning the mathematical studies of Jacob Bernoulli occurred when his younger brother, Johann Bernoulli, began to work on mathematical topics. Johann was told by his father to study medicine but while he was studying that topic he asked his brother Jacob to teach him mathematics.
Gottfried Leibniz1646 - 1716
Ehrenfried Walter von Tschirnhaus
1651 - 1708
Johann1667-1748
Jacob Bernoulli was appointed professor of mathematics in Basel in 1687 and the two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus... published in Acta Eruditorum. They also studied the publications of von Tschirnhaus. It must be understood that Leibniz's publications on the calculus were very obscure to mathematicians of that time and the Bernoullis were the first to try to understand and apply Leibniz's theories.
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Jacob1654-1705
Jacob and Johann Bernoulli
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Jacob.html
Johann1667-1748
Although Jacob and Johann both worked on similar problems their relationship was soon to change from one of collaborators to one of rivals. Johann Bernoulli’s boasts were the first cause of Jacob's attacks on him and Jacob wrote that Johann was his pupil whose only achievements were to repeat what his teacher had taught him. Of course this was a grossly unfair statement. Jacob continued to attack his brother in print in a disgraceful and unnecessary fashion, particularly after 1697. However he did not reserve public criticism for his brother. He was critical of the university authorities at Basel and again he was very public in making critical statements that, as one would expect, left him in a difficult situation at the university. Jacob probably felt that Johann was the more powerful mathematician of the two and, this hurt since Jacob's nature meant that he always had to feel that he was winning praise from all sides. Hofmann writes:-
Sensitivity, irritability, a mutual passion for criticism, and an exaggerated need for recognition alienated the brothers, of whom Jacob had the slower but deeper intellect.
As suggested by this quote the brothers were equally at fault in their quarrel. Johann would have liked the chair of mathematics at Basel which Jacob held and he certainly resented having to move to Holland in 1695. This was another factor in the complete breakdown of relations in 1697.
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Mathematical Works of Jacob Bernoulli
1. In 1682 Jacob Bernoulli did research work on theory of gravitation, comets, etc.
2. Since 1683, he was a regular contributor to “Journal des Sçavans” and “Acta Eruditorium”, where he published many new theorems in algebra..
3. He was between the first contributors to the theory of probability. His first paper in probability was published in 1685. “Ars conjectandi”, Jacob Bernoulli’s book in theory of probability was published in 1713, after his death.
4. In 1689, he published research papers in the theory of infinite series, where he showed that the series 1/12+1/22+1/32+… converges.5. In 1690, he introduced for the first time the term “integral” in a paper published in “Acta eruditorum” (instead of “Calculus summatorium” used by Leibniz).
6. In 1692, he found out the evolutes of parabola and of the logarithmic spiral.
7. In 1696, Jacob invented the method of solving the differential equation of the form dy/dx + P y = Q yn (Bernoulli’s form), where P and Q are functions of x or constants. He used this to solve geometrical and mechanical problems.
Jacob1654-1705
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Mathematical Works of Jacob Bernoulli (continue)
8. He introduced the idea of polar coordinates in analytical geometry and used it to find the properties of spiral shaped curves.
9. He developed the binomial expansion of (1+1/n)n to show that it converges to a number between 2 and 3 (actual limit is e = 2.73).
11. Jacob found the formulae for finding the radius of curvature in both Cartesian and polar coordinates. .
10. Jacob Bernoulli studied the curve “Lemniscate of Bernoulli” r2=a2 cos 2q
Jacob1654-1705
Jacob Bernoulli
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14Jacob Bernoulli’s “Ars Conjectandi”
James is best known for the work Ars Conjectandi (The Art of Conjecture), published eight years after his death in1713 by his nephew Nicholas. In this work, he described the known results in probability theory and in enumeration, often proving alternative proofs of known results. This work also includes the application of probability theory to games of chance and his introduction of the theorem known as the law of large numbers. The terms Bernoulli trial and Bernoulli numbers result from this work. Bernoulli crater, on the Moon, is also named after him jointly with his brother Johann.
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SOLO Review of Probability
Bernoulli Trials – The Binomial Distribution
knkknk ppk
npp
knk
nnkp
11
!!
!,
JacobBernoulli1654-1705
!
,1
!;;
00 k
k
i
eipkP
k
i
ik
i
pnxE
Probability Density Functions
Cumulative Distribution Function
Mean Value
Variance ppnxVar 1
x
a dtttxa0
1 exp,γ is the incomplete gamma function
Moment Generating Function
npjp 1exp DistributionExamples
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SOLO Review of Probability
Bernoulli Trials – The Binomial Distribution (continue – 1)
p – probability of success (r = 1) of a given discrete trial
q – probability of failure (r=0) of the given discrete trial
1 qp
n – number of independent trials
nkp , – probability of k successes in n independent trials (Bernoulli Trials)
knkknk ppk
npp
knk
nnkp
11
!!
!,
Using the binomial theorem we obtain
n
k
knkn ppk
nqp
0
11
therefore the previous distribution is called binomial distribution.
JacobBernoulli1654-1705
Given a random event r = {0,1}
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 k
nkP ,
The probability of k successful trials from n independent trials is given by
The number of k successful trials from n independent trials is given by
!!
!
knk
n
k
n
with probability knk pp 1 to permutations
and Combinations
DistributionExamples
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Jacob1654-1705
The Bernoulli numbers are among the most interesting and important number sequences in mathematics. They first appeared in the posthumous work "Ars Conjectandi" (1713) by Jakob Bernoulli (1654-1705) in connection with sums of powers of consecutive integers (see Bernoulli (1713) or D.E. Smith (1959)). Bernoulli numbers are particularly important in number theory, especially in connection with Fermat's last theorem (see, e.g., Ribenboim (1979)). They also appear in the calculus of finite differences (Nörlund (1924)), in combinatorics (Comtet (1970, 1974)), and in other fields.
Definitions and main properties of Bernoulli numbers can be found in a great number of articles and books listed in this bibliography. Good introductions are given, e.g., in Ireland and Rosen (1982, 1990), Rademacher (1973), and Nörlund (1924). A handy collection of formulas is in Abramowitz and Stegun (1964). Some older books are entirely devoted to Bernoulli numbers; among them are Chistyakov (1895), Nielsen (1923), and Saalschütz (1893). One should, however, be aware of possible differences in notation and indexing, especially in older publications.
Bernoulli Numbers The Bernoulli numbers Bn play an important role in several topics of mathematics. These numbers can be defined by the power series
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SOLO Bernoulli Polynomials
JacobBernoulli1654-1705
0
1
22
3 23
4 3 24
5 4 35
6 5 4 26
1
1
21
63 1
2 21
230
5 5 1
2 3 65 1 1
32 2 42
B x
B x x
B x x x
B x x x x
B x x x x
B x x x x x
B x x x x x
When evaluated at zero, these definitions correspond to the Bernoulli numbers,
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In polar form:
Differential Geometry in the 3D Euclidean Space
Planar Curves
Lemniscate of Bernoulli
The lemniscate, also called the lemniscate of Bernoulli, is a polar curve whose most common form is the locus of points the product of whose distances from two fixed points (called the foci) a distance 2c away is the constant c2. This gives the Cartesian equation:
222222 2 yxcyx
http://mathworld.wolfram.com/Lemniscate.html
2cos2 22 cr
Jakob Bernoulli published an article in Acta Eruditorum in 1694 in which he called this curve the lemniscus (Latin for "a pendant ribbon"). Bernoulli was not aware that the curve he was describing was a special case of Cassini Ovals which had been described by Cassini in 1680. The general properties of the lemniscate were discovered by G. Fagnano in 1750 (MacTutor Archive)
Jacob Bernoulli1654-1705
The most general form of the lemniscate is a toric section of a torus.
http://www-groups.dcs.st-and.ac.uk/~history/Curves/Lemniscate.html
42222 cycxycx
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Lemniscate of James Bernoulli
There are several methods for drawing a lemniscate. The easiest is illustrated above. Draw a circle and then extend a diameter to become a secant. The center of the lemniscate O will be 2 times the radius of the circle. Through O draw several segments cutting the circle. The pattern of the lemniscate emerges in the first quadrant.
For the more mechanically minded, we suggest using the method described in E. H. Lockwoods' book.
http://curvebank.calstatela.edu/lemniscate/lemniscate.htm Run This
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SOLODifferential Geometry in the 3D Euclidean Space
Planar Curves
Equiangular Spiral
Equiangular spiral (also known as logarithmic spiral, Bernoulli spiral, and logistique) describe a family of spirals. It is defined as a curve that cuts all radii vectors at a constant angle.
The famous Equiangular Spiral was discovered by Descartes, its properties of self-reproduction by James (Jacob) Bernoulli (1654-1705) who requested that the curve be engraved upon his tomb with the phrase "Eadem mutata resurgo" ("I shall arise the same, though changed.")
http://xahlee.org/SpecialPlaneCurves_dir/EquiangularSpiral_dir/equiangularSpiral.html
Logarithmic Spiral
bear http://en.wikipedia.org/wiki/Logarithmic_spiral
Nautilus Shells
René Descartes1596 - 1650
Jacob Bernoulli1654-1705
http://mathworld.wolfram.com/LogarithmicSpiral.html
22Jacob Bernoulli’s Tomb
Bernoulli chose a figure of a logarithmic spiral and the motto "Eadem mutata resurgo" ("Changed and yet the same, I rise again") for his gravestone; the spiral executed by the stonemasons was, however, an Archimedian spiral.
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Johann Bernoulli
Johann Bernoulli1667-1748
Johann Bernoulli was the tenth child of Nicolaus and Margaretha Bernoulli. He was the brother of Jacob Bernoulli but Johann was twelve years younger than his brother Jacob which meant that Jacob was already a young man while Johann was still a child. The two brothers were to have an important influence on each others mathematical development and it was particularly true that in his early years Johann must have been greatly influenced by seeing Jacob head towards a mathematical career despite the objections of his parents.
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Johann.html
Nicolaus and Margaretha Bernoulli tried to set Johann on the road to a business career but, despite his father's strong pushing, Johann seemed to be totally unsuited to a future in business. Johann's father had intended him to take over the family spice business and in 1682, when he was 15 years old, Johann worked in the spice trade for a year but, not liking the work, he did not do well. It was with great reluctance that Johann's father agreed in 1683 to Johann entering the University of Basel. The subject that Johann Bernoulli was to study at university was medicine, a topic that many members of the Bernoulli family ended up studying despite their liking for mathematics and mathematical physics.
At Basel University Johann took courses in medicine but he studied mathematics with his brother Jacob. Jacob was lecturing on experimental physics at the University of Basel when Johann entered the university and it soon became clear that Johann's time was mostly devoted to studying Leibniz's papers on the calculus with his brother Jacob. After two years of studying together Johann became the equal of his brother in mathematical skill.
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SOLOJohann Bernoulli (continue – 1)
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Johann.html
In 1691 Johann went to Geneva where he lectured on the differential calculus. From Geneva, Johann made his way to Paris and there he met mathematicians in Malebranche's circle, where the focus of French mathematics was at that time. There Johann met de l’Hôpital and they engaged in deep mathematical conversations. Contrary to what is commonly said these days, de l’Hôpital was a fine mathematician, perhaps the best mathematician in Paris at that time, although he was not quite in the same class as Johann Bernoulli.
Guillaume François Antoine Marquis de L'Hôpital
1661 - 1704
De l’Hôpital was delighted to discover that Johann Bernoulli understood the new calculus methods that Leibniz had just published and he asked Johann to teach him these methods. This Johann agreed to do and the lessons were taught both in Paris and also at de l’Hôpital 's country house at Oucques. Bernoulli received generous payment from de l’Hôpital for these lessons, and indeed they were worth a lot for few other people would have been able to have given them. After Bernoulli returned to Basel he still continued his calculus lessons by correspondence, and this did not come cheap for de l’Hôpital who paid Bernoulli half a professor's salary for the instruction. However it did assure de l’Hôpital of a place in the history of mathematics since he published the first calculus book “Analyse des infiniment petits pour l'intelligence des lignes courbes” (1696) which was based on the lessons that Johann Bernoulli sent to him.
Gottfried Wilhelmvon Leibniz(1646-1716)
Nicolas Malebranche1638 - 1715
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Guillaume François Antoine Marquis de L'Hôpital
1661 - 1704
Johann Bernoulli (continue – 2)
Johann Bernoulli1667-1748
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Johann.html
De l’Hôpital published the first calculus book “Analyse des infiniment petits pour l'intelligence des lignes courbes” (1696) which was based on the lessons that Johann Bernoulli sent to him.
The well known de l’Hôpital 's rule is contained in this calculus book and it is therefore a result of Johann Bernoulli. In fact proof that the work was due to Bernoulli was not obtained until 1922 when a copy of Johann Bernoulli's course made by his nephew Nicolaus (I) Bernoulli was found in Basel. Bernoulli's course is virtually identical with de l’Hôpital 's book but it is worth pointing out that de l’Hôpital had corrected a number of errors such as Bernoulli's mistaken belief that the integral of 1/x is finite. After de l’Hôpital's death in 1704 Bernoulli protested strongly that he was the author of de l’Hôpital's calculus book. It appears that the handsome payment de l’Hôpital e made to Bernoulli carried with it conditions which prevented him speaking out earlier. However, few believed Johann Bernoulli until the proofs discovered in 1922.
0
0 0
0
lim 0lim lim
lim 0
x x
x x x x
x x
d f xf x d xf x
If theng x d g xg x
d x
De l’Hôpital’s rule
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SOLOJohann Bernoulli (continue – 3)
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Johann.html
Johann Bernoulli had already solved the problem of the catenary which had been posed by his brother in 1691. He had solved this in the same year that his brother posed the problem and it was his first important mathematical result produced independently of his brother, although it used ideas that Jacob had given when he posed the problem. At this stage Johann and Jacob were learning much from each other in a reasonably friendly rivalry which, a few years later, would descend into open hostility. For example they worked together on caustic curves during 1692-93 although they did not publish the work jointly. Even at this stage the rivalry was too severe to allow joint publications and they would never publish joint work at any time despite working on similar topics.
A catenary is the curve in which a heavy uniform chain hangs. Its equation is: 1
cosh2
x xy x e e
Johann Bernoulli1667-1748
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SOLOJohann Bernoulli (continue – 4)
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Johann.html
Johann Bernoulli1667-1748
A stream of mathematical ideas continued to flow from Johann Bernoulli .In 1694 he considered the function y = xx and he also investigated series using the method of integration by parts. Integration to Bernoulli was simply viewed as the inverse operation to differentiation and with this approach he had great success in integrating differential equations. He summed series, and discovered addition theorems for trigonometric and hyperbolic functions using the differential equations they satisfy. This outstanding contribution to mathematics reaped its reward in 1695 when he received two offers of chairs. He was offered a chair at Halle and the chair of mathematics at Groningen. This latter chair was offered to Johann Bernoulli on the advice of Huygens and it was this post which Johann accepted with great pleasure, not least because he now had equal status to his brother Jacob who was rapidly becoming extremely jealous of Johann's progress. The fault was not all on Jacob's side however, and Johann was equally to blame for the deteriorating relations. It is interesting to note that Johann was appointed to the chair of mathematics but his letter of appointment mentions his medical skills and offered him the chance to practice medicine while in Groningen .
Johann Bernoulli had married Drothea Falkner and their first child was seven months old when the family departed for Holland on 1 September 1695. This first child was Nicolaus (I) Bernoulli who also went on to become a mathematician. Perhaps this is a good time to note that two other of Johann's children went on to become mathematicians, Daniel Bernoulli, who was born while the family was in Groningen, and Johann (II) Bernoulli .
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HISTORY OF CALCULUS OF VARIATIONS
x
y
V
t
fcfc yxB ,fcx
fcy
N
0,0A
The brachistochrone problem
In 1696 proposed the Brachistochrone (“shortest time”) Problem:Given two points A and B in the vertical plane, what is the curve traced by a point acted only by gravity, which starts at A and reaches B in the shortest time.
Johann Bernoulli 1667 - 1748
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A
B
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H.J. Sussmann, J.C. Willems“300 Years of Optimal Control:From the Brachystochrone to theMaximal Principle”,IEEE Control Systems, June 1997,pp. 32 - 44
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HISTORY OF CALCULUS OF VARIATIONS
The brachistochrone problem
sinr
cosr
r
x
y
0x
0V
g
V
2
20
r
rA
B
),( yx
g
Vry
rxx
2cos1
sin2
0
0
Cycloid Equation
cfcfcf xxxt
xdxd
ydyxFxd
ygV
xd
yd
V
sdtdJ
002
0
2
00
,,2
1
Minimization Problem
Solution of the Brachistochrone Problem:
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HISTORY OF CALCULUS OF VARIATIONSThe brachistochrone problem
Jacob Bernoulli(1654-1705)
Gottfried Wilhelmvon Leibniz(1646-1716)
Isaac Newton(1643-1727)
The solutions of Leibniz, Johann Bernoulli, Jacob Bernoulliand Newton were published on May 1697 publication ofActa Eruditorum. L’Hôpital solution was published only in 1988.
Guillaume FrançoisAntoine de L’Hôpital
(1661-1704)
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SOLOJohan Bernoulli (continue – 5)
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Johann.html
Johann Bernoulli1667-1748
In 1705 return to Basel Johann worked hard to ensure that he succeeded to his brother's chair and soon he was appointed to Jacob's chair of mathematics. It is worth remarking that Bernoulli's father-in-law lived for three years in which he greatly enjoyed having his daughter and grandchildren back in Basel. There were other offers that Johann turned down, such as Leiden, a second offer from Utrecht and a generous offer for him to return to Groningen in 1717.
In 1713 Johann became involved in the Newton-Leibniz controversy. He strongly supported Leibniz and added weight to the argument by showing the power of his calculus in solving certain problems which Newton had failed to solve with his methods. Although Bernoulli was essentially correct in his support of the superior calculus methods of Leibniz, he also supported Descartes' vortex theory over Newton's theory of gravitation and here he was certainly incorrect. His support in fact delayed acceptance of Newton's physics on the Continent.
Bernoulli also made important contributions to mechanics with his work on kinetic energy, which, not surprisingly, was another topic on which mathematicians argued over for many years. His work Hydraulica is another sign of his jealous nature. The work is dated 1732 but this is incorrect and was an attempt by Johann to obtain priority over his own son Daniel. Daniel Bernoulli completed his most important work Hydrodynamica in 1734 and published it in 1738 at about the same time as Johann published Hydraulica. This was not an isolated incident, and as he had competed with his brother, he now competed with his own son. As a study of the historical records has justified Johann's claims to be the author of de l'Hôpital's calculus book, so it has shown that his claims to have published Hydraulica before his son wrote Hydrodynamica are false.
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http://curvebank.calstatela.edu/lemniscate
Johann BernoulliJohann Bernoulli
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SOLODifferential Geometry in the 3D Euclidean Space
Planar Curves
Astroid
The Astroid was studied by Johan Bernoulli (1667 – 1748) ,by D’Alembert in 1748. The name was given by Littrow in 1838.
Johann Bernoulli1667-1748
Jean Le Rond D’Alembert1717 - 1783
3
3
sin
cos
ay
ax
The Astroid can be obtained as a hypocycloid with b/a=1/4 or 3/4
The astroid is the curve performed bya bus door
The astroid is the curve obtained by the intersection of two circles rolling without slipping inside a bigger circle. Run This
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Daniel Bernoulli
Daniel Bernoulli1700 - 1782
1700 - Born in Groningen, the son of Johann Bernoulli, nephew of Jakob Bernoulli, younger brother of Nicolaus II Bernoulli, and older brother of Johann II,
1713 - Daniel was sent to Basel University at the age of 13 to study philosophy and logic.
1715 - He obtained his baccalaureate examinations.
1716 - He obtained his master's degree . During the time he studied philosophy at Basel, he was learning the methods of the calculus from his father and his older brother Nicolaus(II) Bernoulli.
1718/9 - studied medicine at Heidelberg and Strasbourg
1720 - He returned to Basel to complete his doctorate in medicine. By this stage Johann Bernoulli was prepared to teach his son more mathematics while he studied medicine and Daniel studied his father's theories of kinetic energy. What he learned on the conservation of energy from his father he applied to his medical studies and Daniel wrote his doctoral dissertation on the mechanics of breathing. So like his father Daniel had applied mathematical physics to medicine in order to obtain his medical doctorate.
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Daniel.html
1720-25 Venice Italy where he worked on mathematics
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Daniel Bernoulli
Daniel Bernoulli1700 - 1782
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Daniel.html
1724 - Mathematical Exercises was published, with Goldbach's assistance,. This consisted of four separate partsThe first part described the game of faro and is of little importance other than showing that Daniel was learning about probability at this time. The second part was on the flow of water from a hole in a container and discussed Newton's theories (which were incorrect). Daniel had not solved the problem of pressure by this time but again the work shows that his interest was moving in this direction. His medical work on the flow of blood and blood pressure also gave him an interest in fluid flow. The third part of Mathematical exercises was on the Riccati differential equation while the final part was on a geometry question concerning figures bounded by two arcs of a circle.
1725 Return to Basel from Venice. In Venice, Daniel had also designedan hour glass to be used at sea so that the trickle of sand was
constant even when the ship was rolling in heavy seas He submitted his work on this to the Paris Academy and in 1725
won the first prize.Daniel had also attained fame through his work Mathematical
exercises and on the strength of this he was invited to take up the chair of mathematics at St Petersburg. His brother Nicolaus(II) Bernoulli was also offered a chair of mathematics at St Petersburg so in late 1725 the two brothers traveled to St Petersburg. Within eight months of their taking up
the appointments in St Petersburg Daniel's brother died of fever.
Nicolaus II1695-1720
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Daniel Bernoulli
Daniel Bernoulli1700 - 1782
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Daniel.html
1725 - 33 Daniel Bernoulli stayed at St Petersburg.
Leonhard Euler1707 - 1783
1727 - Leonard Euler a pupil of his father Johann Bernoulli joined him at St Petersburg.
1728 - Bernoulli and Euler dominated the mechanics of flexible and elastic bodies, in that year deriving the equilibrium curves for these bodies
00
0 & 0x
x
uu
x
(fixed end)
2 3
2 30 & 0
x L x L
u u
x x
(free end)
2
2
uEI F
x x
Fixed end Free end
ux
F
http://en.wikipedia.org/wiki/Euler-Bernoulli_beam_equation
Euler – Bernoulli beam
While in St Petersburg he made one of his most famous discoveries when he defined the simple nodes and the frequencies of oscillation of a system. He showed that the movements of strings of musical instruments are composed of an infinite number of harmonic vibrations all superimposed on the string.
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Daniel Bernoulli
Daniel Bernoulli1700 - 1782
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Daniel.html
1725 - 33 Daniel Bernoulli stayed at St Petersburg.
A second important work which Daniel produced while in St Petersburg was one on probability and political economy. Daniel makes the assumption that the moral value of the increase in a person's wealth is inversely proportional to the amount of that wealth. He then assigns probabilities to the various means that a person has to make money and deduces an expectation of increase in moral expectation. Daniel applied some of his deductions to insurance.
Undoubtedly the most important work which Daniel Bernoulli did while in St Petersburg was his work on hydrodynamics. Even the term itself is based on the title of the work which he produced called Hydrodynamica and, before he left St Petersburg, Daniel left a draft copy of the book with a printer. However the work was not published until 1738 and although he revised it considerably between 1734 and 1738, it is more the presentation that he changed rather then the substance.
2
2
vg h p const
Bernoulli’s Principle
http://en.wikipedia.org/wiki/Bernoulli%27s_principle
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Bernoulli's diagram to illustrate how pressure is measured. See also part of Bernoulli's original Latin explanation. Source: The Turner Collection, Keele University
Daniel Bernoulli (1700-1782)
http://plus.maths.org/issue1/bern/
His chief work is his Hydrodynamique (Hydrodynamica), published in 1738; it resembles Joseph Louis Lagrange's Méchanique Analytique in being arranged so that all the results are consequences of a single principle, namely, conservation of energy. This was followed by a memoir on the theory of the tides, to which, conjointly with the memoirs by Euler and Colin Maclaurin, a prize was awarded by the French Academy: these three memoirs contain all that was done on this subject between the publication of Isaac Newton's Philosophiae Naturalis Principia Mathematica and the investigations of Pierre-Simon Laplace.
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KINETIC THEORY OF GASES
HISTORY
STATE EQUATION )BOYLE-MARIOTTE LAW(
p - PRESSURE (FORCE / SURFACE)
V - VOLUME OF GAS
m - MASS OF GAS
T - GAS TEMPERATURE
-GAS DENSITY
m3
kg
oK
kg m/ 3
2/ mN
ROBERT BOYLE )1660( DISCOVERED EXPERIMENTALLY THAT THEPRODUCT OF PRESSURE AND VOLUME IS CONSTANT FOR A FIXED
MASS OF GAS AT CONSTANT TEMPERATURE
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mTconstVp ,
ROBERT BOYLE )1627-1691(
New Experiments Physio-Mechanicall,Touching the Spring of the Air, andIts Effects )1660(
p
V
constT
41
KINETIC THEORY OF GASES
HISTORY
RICHARD TOWNLEY )1628-1707( AND HENRY POWER )1623-1668( FOUNDALSO EXPERIMENTALLY THE P*V LAW IN 1660.
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STATE EQUATION )BOYLE-MARIOTTE LAW(
EDMÉ MARIOTTE )~1620-1684( INDEPENDENTLY FINDS THERELATIONSHIP BETWEEN PRESSURE AND VOLUME P*V LAW IN 1676
IN HIS WORK “On the Nature of Air.” mTconstVp ,THIS IS KNOWN AS “MARIOTTE’s LAW” IN FRANCE AND “BOYLE’s LAW”ELSEWHERE.
DANIEL BERNOULLI )1700-1782( IN THE TREATISE “Hydrodynamica” )1738( DERIVES THE
BOYLE-MARIOTTE LAW USING A “BILLIARD BALL”MODEL. HE ALSO USES CONSERVATION OF
MECHANICAL ENERGY TO SHOW THAT THE PRESSURECHANGES PROPORTIONALLY TO THE SQUARE OFPARTICLE VELOCITIES AS TEMPERATURE CHANGES.
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Daniel Bernoulli
Daniel Bernoulli1700 - 1782
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Daniel.html
1733 - He left St Petersburg with his younger brother Johann(II) Bernoulli who was also with him in St Petersburg and they, making visits toDanzig, Hamburg, Holland and Paris before returning to Basel in 1734
Johann II1710-1790
1734 - Daniel Bernoulli submitted an entry for the Grand Prize of the Paris Academy for 1734 giving an application of his ideas to astronomy. This had unfortunate consequences since Daniel's father, Johann Bernoulli, also entered for the prize and their entries were declared joint winners of the Grand Prize. The result of this episode of the prize of the Paris Academy had unhappy consequences for Daniel. His father was furious to think that his son had been rated as his equal and this resulted in a breakdown in relationships between the two. The outcome was that Daniel found himself back in Basel but banned from his father's house. Whether this caused Daniel to become less interested in mathematics or whether it was the fact that his academic position was a non mathematical one, certainly Daniel never regained the vigour for mathematical research that he showed in St Petersburg.
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Daniel Bernoulli
Daniel Bernoulli1700 - 1782
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Daniel.html
1733 - Although Daniel had left St Petersburg, he began an immediate correspondence with Euler and the two exchanged many ideas on vibrating systems. Euler used his great analytic skills to put many of Daniel's physical insights into a rigorous mathematical form. Daniel continued to work on polishing his masterpiece Hydrodynamica for publication and added a chapter on the force of reaction of a jet of fluid and the force of a jet of water on an inclined plane. In this chapter, Chapter 13, he also discussed applications to the propulsion of ships.
1737 - The prize of the Paris Academy also had a nautical theme, the best shape for a ship's anchor, and Daniel Bernoulli was again the joint winner of this prize, this time jointly with Poleni
Botany lectures were not what Daniel wanted and things became better for him in 1743 when he was able to exchange these for physiology lectures. In 1750, however, he was appointed to the chair of physics and taught physics at Basel for 26 years until 1776. He gave some remarkable physics lectures with experiments performed during the lectures. Based on experimental evidence he was able to conjecture certain laws which were not verified until many years later. Among these was Coulomb's law in electrostatics.
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Daniel Bernoulli
Daniel Bernoulli1700 - 1782
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Daniel.html
Daniel Bernoulli did produce other excellent scientific work during these many years back in Basel. In total he won the Grand Prize of the Paris Academy 10 times, for topics in astronomy and nautical topics.
1740 - won (jointly with Euler) the Grand Prize of the Paris Academy for work on Newton's theory of the tides
1743 - won the Grand Prize of the Paris Academy for essays on magnetism
1746 - won the Grand Prize of the Paris Academy for essays on magnetism
1747 - won the Grand Prize of the Paris Academy for a method to determine time at sea
1751 - won the Grand Prize of the Paris Academy for an essay on ocean currents
1753 - won the Grand Prize of the Paris Academy for the effects of forces on ships
1757 - won the Grand Prize of the Paris Academy for proposals to reduce the pitching and tossing of a ship in high seas.
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Daniel Bernoulli
Daniel Bernoulli1700 - 1782
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Daniel.html
Another important aspect of Daniel Bernoulli's work that proved important in the development of mathematical physics was his acceptance of many of Newton's theories and his use of these together with the tolls coming from the more powerful calculus of Leibniz. Daniel worked on mechanics and again used the principle of conservation of energy which gave an integral of Newton's basic equations. He also studied the movement of bodies in a resisting medium using Newton's methods.
Isaac Newton1643 - 1727
He also continued to produce good work on the theory of oscillations and in a paper he gave a beautiful account of the oscillation of air in organ pipes
Daniel Bernoulli was much honoured in his own lifetime. He was elected to most of the leading scientific societies of his day including those in Bologna, St Petersburg, Berlin, Paris, London, Bern, Turin, Zurich and Mannheim.
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Nicolaus II Bernoulli
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Daniel.html
Nicolaus, who showed great promise in the field of mathematics, was called to the St.Petersburg Academy, in 1727, where he unfortunately died of fever, only eight months later. He wrote on curves, differential equations, and probability.
Nicolaus(II) Bernoulli was the favourite of three sons of Johann Bernoulli. He entered the University of Basel when only 13 years of age and, like many other members of his family, studied both mathematics and law. In 1715 he became a licentiate in jurisprudence.
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Nicolaus(II).html
Nicolaus worked as his father's assistant helping him with correspondence. In particular he was involved with writing letters concerning the famous priority dispute between Newton and Leibniz. He not only replied to Taylor regarding the dispute but he also made important mathematical contributions to the problem of trajectories while working on the mathematical arguments behind the dispute.
Nicolaus worked on curves, differential equations and probability. He died only 8 months after taking up an appointment in St Petersburg at a young age when his talents promised so much for the future.
Nicolaus II1695-1720
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Johann II Bernoulli
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Johann(II).html
http://library.thinkquest.org/22584/temh3007.htm
Johann II, the youngest of the three sons, of Johann Bernoulli, studied law but spent his later years as a professor of mathematics at the University of Basel. He was particularly interested in the mathematical theory of heat and light.
Johann II1710-1790He worked on mathematics both with his father and as an independent
worker. He had the remarkable distinction of winning the Prize of the Paris Academy on no less than four separate occasions. On the strength of this he was appointed to his father's chair in Basel when Johann Bernoulli died.
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Nicolaus I Bernoulli
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Nicolaus(I).html
Nicolaus(I) Bernoulli (1687-1759), was a nephew of Jacob Bernoulli and Johann Bernoulli. His early education involved studying mathematics with his uncles. In fact it was Jacob Bernoulli who supervised Nicolaus's Master's degree at the University of Basel which he was awarded in 1704. Five years later he was received a doctorate for a dissertation which studied the application of probability theory to certain legal questions.
Nicolaus Bernoulli was appointed to Galileo's chair at Padua in 1716 which Hermann had filled immediately prior to Nicolaus's appointment. There he worked on geometry and differential equations. In 1722 he left Italy and returned to his home town to take up the chair of logic at the University of Basel. After nine years, remaining at the University of Basel, he was appointed to the chair of law. In addition to these academic appointments, he did four periods as rector of the university.
From Montmort's work we can see that Nicolaus formulated certain problems in the theory of probability, in particular the problem which today is known as the St Petersburg problem. Nicolaus also corresponded with Leibniz during the years 1712 to 1716. In these letters Nicolaus discussed questions of convergence, and showed that (1+x)n diverges for x > 0.
A problem in probability, which he proposed from St. Peresburg, later became known as the Petersburg paradox. The problem is:If A receives a penny when a head appears on the first toss of a coin, two pennies if a head does not appear until second toss, four pennies if a head does not appear until the third toss, and so on, what is A's expectation? Mathematical theory shows that A's expectation is infinite, which seems a paradoxical result. The problem was incestigated by Nicolaus' brother Daniel, who succeeded Nicolaus at St. Petersburg.
http://library.thinkquest.org/22584/temh3007.htm http://en.wikipedia.org/wiki/St._Petersburg_Paradox
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Nicolaus I Bernoulli
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Nicolaus(I).html
In his letters to Euler (1742-43) he criticises Euler's indiscriminate use of divergent series. In this correspondence he also solved the problem of the sum of the reciprocal squares ∑ (1/n2) = π2/6, which had confounded Leibniz and Jacob Bernoulli.
Nicolaus Bernoulli assisted in the publication of Jacob Bernoulli's Ars conjectandi. Later Nicolaus edited Jacob Bernoulli's complete works and supplemented it with results taken from Jacob's diary. Other problems he worked on involved differential equations. He studied the problem of orthogonal trajectories, making important contributions by the construction of orthogonal trajectories to families of curves, and he proved the equality of mixed second-order partial derivatives. He also made significant contributions in studying the Riccati equation.
One of the great controversies of the time was the Newton Leibniz argument. As might be expected Nicolaus supported Leibniz but he did produce some good arguments in his favour such as observing that Newton failed to understand higher derivatives properly which had led him into errors in the problem of inverse central force in a resisting medium.
Nicolaus(I) Bernoulli received many honours for his work. For example he was elected a member of the Berlin Academy in 1713, a Fellow of the Royal Society of London in 1714, and a member of the Academy of Bologna in 1724.
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Johann III Bernoulli
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Johann(III).html
http://library.thinkquest.org/22584/temh3007.htm
Johann Bernoulli II had a son Johann III (1744-1807) who, like his father, studied law but then turned to mathematics. When barely nineteen years old, he was called as a professor of mathematics to the Berlin Academy. He wrote on astonomy, the doctrine of chance, recurring decimals, and indeterminate equations..
Johann III1744-1807
In the field of mathematics he worked on probability, recurring decimals and the theory of equations. As in his astronomical work there was little of lasting importance. He did, however, publish the Leipzig Journal for Pure and Applied Mathematics between 1776 and 1789.
He was well aware of the famous mathematical line from which he was descended and he looked after the wealth of mathematical writings that had passed between members of the family. He sold the letters to the Stockholm Academy where they remained forgotten about until 1877. At that time when these treasures were examined, 2800 letters written by Johann(III) Bernoulli himself were found in the collection.
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Jacob II Bernoulli
http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Jacob(II).html
Jacob (II) Bernoulli was one of the sons of Johann(II) Bernoulli. Following the family tradition he took a degree in law but his interests were in mathematics and mathematical physics.
In 1782 Jacob(II) Bernoulli's uncle Daniel Bernoulli died and his chair of physics in Basel became vacant. Jacob(II) applied for the chair and presented a work on mathematical physics to support his application. The decison as to who should fill the vacant chair was not made on academic grounds but was made by drawing lots. Jacob(II) Bernoulli was unlucky and he was not offered this position he would really have liked.
He was then appointed as secretary to the Imperial Envoy to Turin and Venice. However, he was soon given the chance of another academic post when he received an offer from St Petersburg. He went to St Petersburg and began to write important works on mathematical physics which he presented to the St Petersburg Academy of Sciences. These treatises were on elasticity, hydrostatics and ballistics.
Despite the rather harsh climate, the city of St Petersburg had great attractions for Jacob(II) Bernoulli since his uncle Daniel Bernoulli had worked there with Euler. In fact Jacob(II) married a granddaughter of Euler in St Petersburg but, tragically, the city was to lead to his death.
Jacob II1759-1789
http://www.ub.unibas.ch/spez/bernoull.htm
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http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Daniel.html
http://library.thinkquest.org/22584/temh3007.htm
Lesser Bernoulli descendants are Daniel II (1751-1834) and Jakob II (1759-1789), two other sons of Johann II, Christoph (1782-1834), a son of Daniel II, and Johann Gustav (1811-1863), a son of Christoph.
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Burial-place of Johann III Bernoulli in Berlin-Köpenick http://www.w-volk.de/museum/grave05.htm
Johann III Bernoulli was buried on the cemetery of the St. Laurentius Church in Berlin-Köpenick. It seems that the grave is not longer existant but the metallic cross with the names and the dates of him and his wife is still there.
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http://www.math.wichita.edu/history/men/bernoulli.html
http://www.maths.tcd.ie/pub/HistMath/People/Bernoullis/RouseBall/RB_Bernoullis.html
http://library.thinkquest.org/27694/Bernoulli%20Family.htm
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Jacob.html
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Johann.html
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Daniel.html
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Johann(II).html
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bernoulli_Nicolaus(I).html
http://www.bookrags.com/biography/jakob-bernoulli/
http://www.fluidmech.net/tutorials/bernoulli/bernoulli-history.htm
http://www.roma.unisa.edu.au/07305/fhome.htm
http://en.wikipedia.org/wiki/Bernoulli%27s_principle
http://en.wikipedia.org/wiki/Euler-Bernoulli_beam_equation
http://en.wikipedia.org/wiki/Daniel_Bernoulli
http://www.fyma.ucl.ac.be/~gaino/Bernoulli/aboutus.html
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TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 – 2013
Stanford University1983 – 1986 PhD AA