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2.  Laplace and Lagrange were visitors at the Cauchy family
 In1805 he took the entrance examination for the École Polytechnique. He was
examined by Biota and placed second. At the École Polytechnique his analysis tutor
was Ampère.
 In 1807 he graduated from the École Polytechnique and entered the engineering
school École des Ponts et Chaussées.
 In 1810 Cauchy took up his first job in Cherbourg to work on port facilities for
Napoleon's English invasion fleet
 In 1816 he won the Grand Prix of the French Academy of Science for a work on
waves. He achieved real fame however when he submitted a paper to the Institute
solving one of Fermat's claims on polygonal numbers made to Marlene.
 1817 lectured on methods of integration at the Collège de France.
 His text Cours d'analyse in 1821 was designed for students at École Polytechnique
and was concerned with developing the basic theorems of the calculus as rigorously
as possible.
 In 1831 Cauchy went to Turin and after some time there he accepted an offer from
the King of Piedmont of a chair of theoretical physics.
 In 1833 Cauchy went from Turin to Prague in order to follow Charles X and to tutor
his grandson. Met with Bolzano.
 Cauchy returned to Paris in 1838 and regained his position at the Academy
 Numerous terms in mathematics bear Cauchy's name:- the Cauchy integral theorem,
in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for
the solution of partial differential equations, the Cauchy-Riemann equations and
Cauchy sequences. He produced 789 mathematics papers,

3.  First Lesson: Introduces the notions of limits and defines
infinitesimals in terms of limits. An infinitesimal variable is
considered to be a sequence whose limit is zero.
 Second Lesson: Definition of continuity
f(x+i)-f(x) is infinitesimal
 Third Lesson: Definition of derivative:
 Twenty-First Lesson: Definition of integration
x = f ( x + i ) - f ( x
)
D
i
D
y
◦ Partition [x0,X] into [x0,x1], … ,[xn-1,X]
◦ Sum: S= (x1-x0)f(x0)+(x2-x1)f(x1)+ … + (X-xn-1)f(xn-1)
◦ Take the limit with more and more intermediate values.
Fixing Dx=h=dx rewrite S=S h f(x)=S f(x) Dx which becomes
in the limit. The notation for the bounds is due to Fourier.
The additivity propriety of the integral with respect to the domain is
also given.
ò X
x
f ( x )
dx
0

4.  Twenty-Sixth Lesson: Indefinite integrals are defined and using
the Intermediate Value Theorem for Integrals, it is shown that
F(x) is continuous. X
: = ò
f ( x Moreover )
dx
F(x) is differentiable and
F(x)’(x)=f(x). This is a version x
0
of the Fundamental Theorem.
 Applications:
◦ Q.: Solve w’(x)=0! A.: w(x)=c.
◦ Q.: Solve y’=f(x)! A.:
= ò +
y f x dx x
with w’(x)=0 or y=∫f(x)dx=F(x)+w(x), where F(x) a particular solution.
◦ Set F(x)= then F(x)=F(X)-F(x0)
for any particular solution F of F’(x)=f(x)
x
x
0
( ) w( )
ò X
x
f ( x )
dx
0

5.  Bernhard Riemann (1826 -1866) improved Cauchy’s definition by using the
sums
S= (x1-x0)f(c0)+(x2-x1)f(c1)+ … + (X-xn-1)f(cn-1)
with xi≤ci≤xi+1.
which are now called Riemann sums. With this definition it is possible to
integrate more functions.
 Henri Léon Lebesgue (1875-1941) found a new way to define integrals, with
which it is possible to integrate even more functions. For this one uses so-called
simple functions as an approximation and measures their contribution
by what is called a Lebesgue mesure. This is technically more difficult and
outside the scope of usual calculus classes. It is however the integral of
choice and is used e.g. in quantum mechanics.
 The Lebesgue integral can for instance be used to integrate the function
f(x) defined by Dirichlet which is given by f(x)=1 if x is irrational and f(x)=0 if
x is rational. The answer is 1. Notice that the limit of the Riemann sums
does not exist, however.

6.  Bernhard Riemann (1826 -1866) improved Cauchy’s definition by using the
sums
S= (x1-x0)f(c0)+(x2-x1)f(c1)+ … + (X-xn-1)f(cn-1)
with xi≤ci≤xi+1.
which are now called Riemann sums. With this definition it is possible to
integrate more functions.
 Henri Léon Lebesgue (1875-1941) found a new way to define integrals, with
which it is possible to integrate even more functions. For this one uses so-called
simple functions as an approximation and measures their contribution
by what is called a Lebesgue mesure. This is technically more difficult and
outside the scope of usual calculus classes. It is however the integral of
choice and is used e.g. in quantum mechanics.
 The Lebesgue integral can for instance be used to integrate the function
f(x) defined by Dirichlet which is given by f(x)=1 if x is irrational and f(x)=0 if
x is rational. The answer is 1. Notice that the limit of the Riemann sums
does not exist, however.