Editing Calculus of variations (section)

== Extrema ==

The calculus of variations is concerned with the maxima or minima (collectively called '''extrema''') of functionals. A functional maps [[Function (mathematics)|functions]] to [[scalar (mathematics)|scalars]], so functionals have been described as "functions of functions."  Functionals have extrema with respect to the elements <math>y</math> of a given [[function space]] defined over a given [[Domain of a function|domain]]. A functional <math>J[y]</math> is said to have an extremum at the function <math>f</math> if <math>\Delta J = J[y] - J[f]</math> has the same [[Sign (mathematics)|sign]] for all <math>y</math> in an arbitrarily small neighborhood of <math>f.</math>{{efn|The neighborhood of <math>f</math> is the part of the given function space where <math>|y - f| < h</math> over the whole domain of the functions, with <math>h</math> a positive number that specifies the size of the neighborhood.<ref name='CourHilb1953P169'>{{cite book |last1=Courant |first1=R |author-link1=Richard Courant |last2=Hilbert |first2=D |author-link2=David Hilbert |title = Methods of Mathematical Physics |volume=I |edition=First English |publisher=Interscience Publishers, Inc. |year=1953 |location=New York |page=169 |isbn=978-0471504474}}</ref>}} The function <math>f</math> is called an '''extremal''' function or extremal.{{efn|name=ExtremalVsExtremum| Note the difference between the terms extremal and extremum. An extremal is a function that makes a functional an extremum.}} The extremum <math>J[f]</math> is called a local maximum if <math>\Delta J \leq 0</math> everywhere in an arbitrarily small neighborhood of <math>f,</math> and a local minimum if <math>\Delta J \geq 0</math> there. For a function space of continuous functions, extrema of corresponding functionals are called '''strong extrema''' or '''weak extrema''', depending on whether the first derivatives of the continuous functions are respectively all continuous or not.<ref name='GelfandFominPP12to13'>{{harvnb|Gelfand|Fomin|2000|pp=12–13}}</ref>

Both strong and weak extrema of functionals are for a space of continuous functions but strong extrema have the additional requirement that the first derivatives of the functions in the space be continuous. Thus a strong extremum is also a weak extremum, but the [[Converse (logic)|converse]] may not hold.  Finding strong extrema is more difficult than finding weak extrema.<ref name='GelfandFominP13'>{{harvnb | Gelfand|Fomin| 2000 | p=13 }}</ref> An example of a [[Necessity and sufficiency|necessary condition]] that is used for finding weak extrema is the [[Euler–Lagrange equation]].<ref name='GelfandFominPP14to15'>{{harvnb | Gelfand|Fomin| 2000 | pp=14–15 }}</ref>{{efn|name=SectionVarSuffCond| For a sufficient condition, see section [[#Variations and sufficient condition for a minimum|Variations and sufficient condition for a minimum]].}}