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...Calculus of variations in one studies min max problems which looks for a number or point that minimizes maximizes some quantity the is about looking not but rather function example given two points x y and nd shortest curve graph joining r with makes arclength z p l ydx as small possible here denotes dy pdx q if we let s denote then ds dx more generally any c can look general minimum might exist however does it has to satisfy dierential equation called euler lagrange solve exists theorem suppose subject boundary conditions solution d notation important understand distinction between note three variables usual its partials respect those composed variable namely derivative written dl thus be proof consider u vanishes on endpoints also satises so every f at fact calculate follows su dsl simply ldu yu integrating second expression by parts gives last since udx have shown this must hold all functions vanish hence...