Preface One November , Ric e Universit y hoste d a grou p o f thirt y undergrad - uate mathematic s major s wit h th e purpos e o f introducin g the m t o research mathematics an d graduat e school . Th e principle part o f thi s introduction wa s the serie s of talks an d workshops , whic h al l took u p some ide a o r them e fro m th e calculu s o f variations . Thes e wer e s o successful tha t th e America n Mathematica l Societ y encouraged u s t o present the m t o a wider audience , i n th e for m yo u se e here. The calculu s o f variation s i s a beautifu l subjec t wit h a ric h his - tory an d wit h origin s i n th e minimizatio n problem s o f calculu s (se e Chapter 1) . Although , a s we will discover i n the chapter s below , i t i s now at th e core of many modern mathematical fields, i t does not hav e a well-define d plac e i n mos t undergraduat e mathematic s course s o r curricula. W e hop e tha t thi s smal l volum e wil l nevertheles s giv e th e undergraduate reade r a sense o f its grea t characte r an d importance . An interestin g stor y motivatin g th e calculu s o f variation s come s from Carthag e i n 90 0 BC , lon g before th e discover y o f calculu s b y Newton an d Leibniz . Quee n Dido , a s a result o f a bargaining negoti - ation, obtaine d "a s much lan d a s could b e enclosed b y the ski n o f a n ox." Sh e had th e ox skin cut int o strips a s thin a s practically possibl e and forme d a lon g cor d o f fixed length . I f he r choic e o f lan d ha d been restricte d t o fla t inlan d territory , the n sh e woul d presumabl y have chosen a large circular region . Thi s is because the circle , amon g vn

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