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...Calculus of variations we consider a functional i fonseca and g leoni u x f z dx carnegie mellon university usa wherex isafunctionspace usuallyalp spaceor sobolev type space rd with rn history an open set n d are positive integers the is branch mathe density function matical analysis that studies extrema critical here in what follows points functionals or energies by func stands for matrix valued distributional tional mean mapping from derivative to real numbers vast theory chose highlight some contemporary as one rst questions may be framed pects eld conclude this article within dido s isoperimetric prob mentioning few forefront areas application lem see subsection nd shape driving current research curve prescribed perimeter maximizes area enclosed was phoenician princess who emigrated north africa upon arrival obtained native chief much territory section address fundamental minimiza she could enclose ox hide cut tion problems relevant techniques cal into long strip used it delineate ...