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...Ccsmath calculus on manifolds simon rubinstein salzedo spring introduction these notes are based a course i took from professor martinscharlemanninthespringof thecoursewasdesignedforrst yearccs math majors the primary textbook was michael spivak s recommendedsupplementary text maxwell rosenlicht to analysis chapter basic and topology wedene rn x xn xi r for example e is vector space this means that we have two operations rrn satisfying following properties y z there exists so all if yn dene assume of norm with useful it dened by n p proposition satises equality linearly dependent proof then some j suppose without loss generality yi xiyi xxi...