San José State University |
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applet-magic.comThayer WatkinsSilicon Valley, Tornado Alley & the Gateway to the Rockies USA |
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The Calculus of Variationsin Functionals |
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This field of mathematics should probably be called *The Calculus of Functionals*.
But the interest in this field arose from an interest in extremes and the conditions for
extremes involve variations and thus arose the name.

A functional is just a function from a set of functions on an interval of real numbers to the set of real nmbers. It could be as simple as the maximum of a particular set of functions defined on the interval [0, 1]. More typical are the functionals defined in terms of an integral; e.g.,

Functionals could be defined for more than one set of functions. For example let A and B be two sets of functions. The F: A×B → R would be a binary functional. Not much has bring done with multinary functionals.

Much of the Calculus of Variations involves functionals which are dependent upon the derivative of the function as well as the function itself. For example, Let g(x, y, z) be a function of three variables. Then

is a functional. Functionals are customarily expressed using square brackets [.] instead of parentheses (. ).

- f + g = g + f

(Commutation) - (f + g) + h = f + (g + h)

(Associativity) - There exists an element 0

such that

f + 0 = f

for all f. - For each f there exist an element −f

such that f + (−f) = 0 - 1·f = f
- α·(f + g) = α·f + α·g (Distributivity)
- (α + β)·f = α·f + β·f
- α·(β·f) = (α·β)·f
- ||f|| = 0 if and only if f=0

- ||α·f|| = |α|·||f||
- ||f + g|| ≤ ||f|| + ||g||

Functional can be subjected to fuller analysis if the sets of functions they are defined over constitute a generalization of a vector space. This means that for any f, g and h in the set and α and β real numbers

A function space may be considered a vector space with the dimensionality which is the cardinality of the continuum.

A function space is *normed* if there exists a real valued nonnegative
function ||. || over the set of functions such that

The distance between f and g is ||f −g||= ||f +(−g)||

A functional F[f] is continuous at a point f=f_{0}
if

for any ε>0 there exists a δ>0

such that if ||f−f_{0}|| < δ

then |F[f] − F[f_{0}]| < ε.

- Let δ be a function in a function space and F[f] a functional over
that same space. The variation in F[f] with respect to δ is
#### Δ

_{*δ}F[f] = F[f+δ] −F[f]- A necessary condition for a function f
_{0}to produce an extreme of a functional F[f] is that#### Δ

_{*δ}F[f_{0}] = 0

for all admissible δ - Euler's Equation:

Let F[f] be a functional of the form

#### F[f] = ∫

_{a}^{b}g(x, f(x), f' (x))dxdefined over a set of functions {f} having continuous first derivatives over the interval [a, b] and having the boundary conditions f(a)=A and f(b)=B. A nessary condition for F[f] to have an extreme for f=f

_{0}is that#### (∂F/∂f(x)) − (d(∂F/∂f)/dx)) = 0

evaluated at f_{0}for all x.

## Necessary Conditions for Extrema

(To be continued.)

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