2.2 Fundamental Conceptions of the Calculus of Variations
91
For the symbols F, C and C
n , they can also be applied to multivariate function.
At this time, change the above interval range to the domain that function relies on.
Let F = {y(x)} be a given function class, R is a real number set. If for each
function y(x) of the function class F, a variable J in R has a determined value
corresponding with it according to certain rules, then J is called the functional of
the function y(x) (in the function class F), it is written as J = J [y(x)], J = J [y(·)]
or J = J [y]. The function y(x) is called the argument of the functional J, sometimes
is also called the functional variable or argument function. The function class F
is called the domain of definition or domain of the functional J. The condition
attached to the argument function is called the admissible condition. the argument
function belonging to the domain is called the admissible function. In other words,
the functional is a real-valued function taking a function class as the domain of
definition. The functional is a generalization of the concept of function. In order to
differ from the ordinary function, the function that functional depend on is enclosed
in square brackets. The term functional was posed by Hadamard in 1910 in his book
Leçons sur le calcul des variations, Vol. 1.
It can be seen from the above definition of functional that the value of a functional is a number, while the independent variable that relies on is a function, so a
functional is the corresponding relation of variable and function, it is a kind of generalized function, while a function is the corresponding relation of the variable and the
variable, it is the basic difference between a functional and a function. Moreover, a
composite function relies on the independent variable x, when the value of x is given,
a corresponding value of the composite function can be work out. A functional is
depends on the function y(x), a value of the functional neither depends on a value of
the independent variable x, nor depends on a value of the function y(x), but depends
on the functional relations of y(x) in the function class and x, The functional depends
on the whole function y(x) and a interval of x, This is the basic difference between
a functional and a compound function.
Let F = {y(x)} be a given function class, J is a functional of y(x) in F, k
is an arbitrary constant, if J [ky(x)] = k
n J [y(x)], the J [y(x)] is called the n th
homogeneous functional defined in F.
Because is the geometric figure of the function of one variable is a curve, the
function of one variable can also be called the curve function. Similarly, in general,
the form of expression of the function of two variables in the geometry is a surface,
so the function of two variables can also be called the surface function.
When x is a variable in multi-dimensional domain (x 1 , x 2 , · · · , x n ), the functional of the above definition is also applicable, at this moment the functional is
written as J = J [u(x 1 , x 2 , · · · , x n )]. In the same way, the functional that depends on
multiple unknown functions is written as J = J [y 1 (x), y 2 (x), · · · , y m (x)], where,
y 1 (x), y 2 (x), . . . , y m (x) are all independent functions. Moreover, a functional can be
written as J = J [u 1 (x 1 , x 2 , · · · , x n ), u 2 (x 1 , x 2 , · · · , x n ), · · · , u m (x 1 , x 2 , · · · , x n )],
where, u 1 (x 1 , x 2 , · · · , x n ), u 2 (x 1 , x 2 , · · · , x n ), . . . , u m (x 1 , x 2 , · · · , x n ) are also
independent functions.
For instance, Let the surface S be given by the equation u = u(x, y), the function
u(x, y) has continuous partial derivative, the projection of S in the Oxy plane is D,
91
For the symbols F, C and C
n , they can also be applied to multivariate function.
At this time, change the above interval range to the domain that function relies on.
Let F = {y(x)} be a given function class, R is a real number set. If for each
function y(x) of the function class F, a variable J in R has a determined value
corresponding with it according to certain rules, then J is called the functional of
the function y(x) (in the function class F), it is written as J = J [y(x)], J = J [y(·)]
or J = J [y]. The function y(x) is called the argument of the functional J, sometimes
is also called the functional variable or argument function. The function class F
is called the domain of definition or domain of the functional J. The condition
attached to the argument function is called the admissible condition. the argument
function belonging to the domain is called the admissible function. In other words,
the functional is a real-valued function taking a function class as the domain of
definition. The functional is a generalization of the concept of function. In order to
differ from the ordinary function, the function that functional depend on is enclosed
in square brackets. The term functional was posed by Hadamard in 1910 in his book
Leçons sur le calcul des variations, Vol. 1.
It can be seen from the above definition of functional that the value of a functional is a number, while the independent variable that relies on is a function, so a
functional is the corresponding relation of variable and function, it is a kind of generalized function, while a function is the corresponding relation of the variable and the
variable, it is the basic difference between a functional and a function. Moreover, a
composite function relies on the independent variable x, when the value of x is given,
a corresponding value of the composite function can be work out. A functional is
depends on the function y(x), a value of the functional neither depends on a value of
the independent variable x, nor depends on a value of the function y(x), but depends
on the functional relations of y(x) in the function class and x, The functional depends
on the whole function y(x) and a interval of x, This is the basic difference between
a functional and a compound function.
Let F = {y(x)} be a given function class, J is a functional of y(x) in F, k
is an arbitrary constant, if J [ky(x)] = k
n J [y(x)], the J [y(x)] is called the n th
homogeneous functional defined in F.
Because is the geometric figure of the function of one variable is a curve, the
function of one variable can also be called the curve function. Similarly, in general,
the form of expression of the function of two variables in the geometry is a surface,
so the function of two variables can also be called the surface function.
When x is a variable in multi-dimensional domain (x 1 , x 2 , · · · , x n ), the functional of the above definition is also applicable, at this moment the functional is
written as J = J [u(x 1 , x 2 , · · · , x n )]. In the same way, the functional that depends on
multiple unknown functions is written as J = J [y 1 (x), y 2 (x), · · · , y m (x)], where,
y 1 (x), y 2 (x), . . . , y m (x) are all independent functions. Moreover, a functional can be
written as J = J [u 1 (x 1 , x 2 , · · · , x n ), u 2 (x 1 , x 2 , · · · , x n ), · · · , u m (x 1 , x 2 , · · · , x n )],
where, u 1 (x 1 , x 2 , · · · , x n ), u 2 (x 1 , x 2 , · · · , x n ), . . . , u m (x 1 , x 2 , · · · , x n ) are also
independent functions.
For instance, Let the surface S be given by the equation u = u(x, y), the function
u(x, y) has continuous partial derivative, the projection of S in the Oxy plane is D,