Chapter 1
Preliminaries
A handy tool makes a handy man (The Analects of Confucius, Chapter XV: Wei
Ling Kung). To learn variational methods require some necessary basic knowledge
of mathematics, such as the Taylor series expansions of function of one variable
and function of several variables, the integral with parameter, the vector analysis
and field theory, the coordinate transformation, the fundamental lemmas of variational methods and the basic concept of tensor, and so on. The basic knowledge of
mathematics is briefly introduced in this chapter.
1.1 The Taylor Formulae
1.1.1 Case of a Function of One Variable
Theorem 1.1.1 If a function f (x) has the continuous derivative of order n +1 in a
certain open interval (a, b) at point x 0 , then when x is within the open interval (a, b),
the function f (x) can be expressed as
f (x) = f (x 0 ) + f (x 0 )(x − x 0 ) +
f (x 0 )
2!
(x − x 0 ) 2 + · · · +
f (n) (x 0 )
n!
(x − x 0 ) n + R n (1.1.1)
where
R n =
f
(n+1)
(ξ )
(n + 1)!
(x − x 0 )
n+1
(1.1.2)
where, ξ is a value between x 0 and x.
Theorem 1.1.1 is called the Taylor mean value theorem or Taylor theorem of
a function of one variable. Equation (1.1.1) is called the Taylor formula of f (x)
expansion to nth order at point x 0 by the power of (x −x 0 ) or is called the Tayler series
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5_1
1
Preliminaries
A handy tool makes a handy man (The Analects of Confucius, Chapter XV: Wei
Ling Kung). To learn variational methods require some necessary basic knowledge
of mathematics, such as the Taylor series expansions of function of one variable
and function of several variables, the integral with parameter, the vector analysis
and field theory, the coordinate transformation, the fundamental lemmas of variational methods and the basic concept of tensor, and so on. The basic knowledge of
mathematics is briefly introduced in this chapter.
1.1 The Taylor Formulae
1.1.1 Case of a Function of One Variable
Theorem 1.1.1 If a function f (x) has the continuous derivative of order n +1 in a
certain open interval (a, b) at point x 0 , then when x is within the open interval (a, b),
the function f (x) can be expressed as
f (x) = f (x 0 ) + f (x 0 )(x − x 0 ) +
f (x 0 )
2!
(x − x 0 ) 2 + · · · +
f (n) (x 0 )
n!
(x − x 0 ) n + R n (1.1.1)
where
R n =
f
(n+1)
(ξ )
(n + 1)!
(x − x 0 )
n+1
(1.1.2)
where, ξ is a value between x 0 and x.
Theorem 1.1.1 is called the Taylor mean value theorem or Taylor theorem of
a function of one variable. Equation (1.1.1) is called the Taylor formula of f (x)
expansion to nth order at point x 0 by the power of (x −x 0 ) or is called the Tayler series
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5_1
1