3.6 Higher Order Variations of Functionals
235
δ
2 J =
x 1
x 0
⎛
⎜
⎜
⎜
⎜
⎜
⎝
n
i,k=1
F y i y k δy i δy k +
n
i,k=1
F y i y
k
δy i δy k
+
n
i,k=1
F y
i y k δy
i δy k +
n
i,k=1
F y
i y
k
δy
i δy
k
⎞
⎟
⎟
⎟
⎟
⎟
⎠
dx
=
x 1
x 0
n
i,k=1
F y i y k δy i δy k + 2
n
i,k=1
F y i y
k
δy i δy k
+
n
i,k=1
F y
i y
k
δy
i δy
k
dx
3.7 Introduction to the Famous Scientists
Legendre (Adrien Marie, 1752.9.18–1833.1.9) French mathematician. Born and
died in Paris. Graduated from the Collège Mazarin in Paris in 1770. Served as a
professor of mathematics at the École Militaire in Paris from 1775. His treatise on
projectiles in resistant media won the prize of the Berlin Academy of Sciences in
1782. Was elected an assistant academician of the Paris Academy of Sciences in
1783, two years later promoted to academician, became a member of the Royal
Society in 1787. Became a permanent member of the French Academy of Sciences
in 1795. He with Lagrange and Laplace were collectively called the “three L” in
French mathematical field. Research field mainly involved mathematical analysis,
elementary geometry, number theory and celestial mechanics etc. aspects. One of the
founders of elliptic integral theory, Published Recherches sur la figure des Planètes
(1784), Eléments de géométrie (1794), Essai sur la Théorie des Nombres (1798,
1808, 1830) and Traité des Fonctions Elliptiques (in three volumes 1825, 1826, and
1830) etc. a large number of papers and works. Made significant contributions to
theory of geodesy, spherical trigonometry and least squares method etc. Also had
discussed the higher geometry, mechanics, astronomy and physics etc.
Jacobi (Carl Gustav Jacbo, 1804.12.10–1851.2.18) German mathematician. Born in
Potsdam, died in Berlin. Went to study at the University of Berlin in 1820, obtained
PhD in 1825. Was elected a member of the Berlin Academy of Sciences in 1827.
Served as a professor at the University of Königsberg in 1832, in the same year to
become a member of the Royal Society. Was an Honorary academician or communication academician of the Saint Petersburg Academy of Sciences, Vienna Academy
of Sciences, Paris Academy of Sciences and Madrid Academy of Sciences etc. Was
one of the founders of the theory of elliptic functions, representative work was Fundamenta nova theoriae functionum ellipticarum (1829, 2012). Established the derivative
formula for function determinant, introduced “Jacobian determinant”, and proposed
the role of these determinants in the multiple integral variable substitution and the
solution of partial differential equation. In the number theory, linear algebra, calculus
of variations, differential equations, differential geometry, complex function theory,
mechanics and history of mathematics etc. all had important contribution. Many
235
δ
2 J =
x 1
x 0
⎛
⎜
⎜
⎜
⎜
⎜
⎝
n
i,k=1
F y i y k δy i δy k +
n
i,k=1
F y i y
k
δy i δy k
+
n
i,k=1
F y
i y k δy
i δy k +
n
i,k=1
F y
i y
k
δy
i δy
k
⎞
⎟
⎟
⎟
⎟
⎟
⎠
dx
=
x 1
x 0
n
i,k=1
F y i y k δy i δy k + 2
n
i,k=1
F y i y
k
δy i δy k
+
n
i,k=1
F y
i y
k
δy
i δy
k
dx
3.7 Introduction to the Famous Scientists
Legendre (Adrien Marie, 1752.9.18–1833.1.9) French mathematician. Born and
died in Paris. Graduated from the Collège Mazarin in Paris in 1770. Served as a
professor of mathematics at the École Militaire in Paris from 1775. His treatise on
projectiles in resistant media won the prize of the Berlin Academy of Sciences in
1782. Was elected an assistant academician of the Paris Academy of Sciences in
1783, two years later promoted to academician, became a member of the Royal
Society in 1787. Became a permanent member of the French Academy of Sciences
in 1795. He with Lagrange and Laplace were collectively called the “three L” in
French mathematical field. Research field mainly involved mathematical analysis,
elementary geometry, number theory and celestial mechanics etc. aspects. One of the
founders of elliptic integral theory, Published Recherches sur la figure des Planètes
(1784), Eléments de géométrie (1794), Essai sur la Théorie des Nombres (1798,
1808, 1830) and Traité des Fonctions Elliptiques (in three volumes 1825, 1826, and
1830) etc. a large number of papers and works. Made significant contributions to
theory of geodesy, spherical trigonometry and least squares method etc. Also had
discussed the higher geometry, mechanics, astronomy and physics etc.
Jacobi (Carl Gustav Jacbo, 1804.12.10–1851.2.18) German mathematician. Born in
Potsdam, died in Berlin. Went to study at the University of Berlin in 1820, obtained
PhD in 1825. Was elected a member of the Berlin Academy of Sciences in 1827.
Served as a professor at the University of Königsberg in 1832, in the same year to
become a member of the Royal Society. Was an Honorary academician or communication academician of the Saint Petersburg Academy of Sciences, Vienna Academy
of Sciences, Paris Academy of Sciences and Madrid Academy of Sciences etc. Was
one of the founders of the theory of elliptic functions, representative work was Fundamenta nova theoriae functionum ellipticarum (1829, 2012). Established the derivative
formula for function determinant, introduced “Jacobian determinant”, and proposed
the role of these determinants in the multiple integral variable substitution and the
solution of partial differential equation. In the number theory, linear algebra, calculus
of variations, differential equations, differential geometry, complex function theory,
mechanics and history of mathematics etc. all had important contribution. Many