## Preliminaries

A function space, , is a set topological vector space (a set) whose elements are functions with a common domain. We assume that functions in the space are all differentiable to any order as needed.

A functional is a function .

It should be noted that a functional eats a function and return a real value rather than eating a value of a function and returning a number. So, if is a functional and a function, we write which is a real number. It doesn’t matter what value of is at any ; a functional sees the function , i.e. the *rule*.

**Remark:** A functional is not a compositions of functions like as acts of the value of not on the rule .

**Example:** The integral is a functional; the same for the sum . Note that can be a constant function like or the identity function . We can also write as follows to better denote that a functional eats a function: .

Adding two functions in a function vector space can be interpreted two ways. Let and . Then, can be interpreted as 1) perturbing with , or 2- is a function produced by moving along the direction of in the function space and reaching the function . The functions are the member of the vector space, hence such that the norm is defined based on an inner product in the function space. The term is considered as the unit vector of and its direction can be relatively calculated with respect to when using the space’s inner product.

# Variation of a function

**Definition:** For a functions , and , the term is called a variation of for an arbitrary function .

Let and be two (fixed) functions. Then, the variation of the composition is defined as,

where is a function of . Note that is a function of and on the domain of .

Observing that is a function of for a fixed , the variation of can be linearized for small variations of . Letting and close to zero, we can use the Taylor series and write,

letting lead to,

which is called the linearized or the first variation of the function due to variation in its argument .

Similarly if with and , then,

**Lemma 1:** If and , then . Proof is as follows.

**Lemma 2:** If , then . Because

# Variation of a functional

Variation of a functional should show its instant variation when there is a infinitesimal change in its argument, i.e. a function.

**Definition: **For a functional , the term is called the variation of due to/in the direction of the variation of .

To evaluate , we observed that is a function of for a fixed and , i.e. for a fixed and is now a real valued function on .

It should be noted that is a function of when and are fixed. Therefore, this rule, , is not a functional anymore; instead, it is a function. The cause of change in the argument of the function, , is the change in . Hence, we can write the following limit (if it exists).

With that limit defined, the Taylor expansion of the function about zero is,

or with keeping in mind that and are fixed and not functions of , above can be written as,

With this regard, the variation of the functional for small becomes,

(1)

which is called the **first variation** of the functional or the Gateaux derivative of the functional. The function is referred to as a **test function**.

The formula for the variation of a functional cannot extended as partial derivatives as in Eq. 1 because for is not defined in the function space (it’s nonsense). However, the variation of a functional in the form of an integral (or sum) operator leads to the variation of the function inside the operator. In other word, the variation operator gets inside the integral operator.

**Example:** let , where , , , be a functional. Then, the variation of is as follows ().

As noted in the example the variation of the functional is now transferred to the variations of functions.

# The basic problem of variational calculus

Let , i.e. a real-valued functional, then the basic problem of variational calculus is expressed as finding for which attains a minimum (or attains a maximum), i.e.

To solve this problem, a necessary condition can be established. Let be a minimum of and be another function. Defining , we can write

For all , when fixed, the term becomes a function of epsilon. This function has a minimum at , because . This means

.

Because is an arbitrary function and is called a variation of (by the definition), we can write,

**Proposition:** If a functional attains a minimum at , then the variation of at is zero for all variations of . In notations,

**Definition:** If for a functional , the such that is called an stationary point of the functional. means,

The proposition says that if minimizes a functional, then it is an stationary point as well.