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1、Annexe3RECOMMENDEDKNOWLEDGEINMATHEMATICS,PHYSICS,CHEMISTRYMATHEMATICS1-ALGEBRA1.1SettheoryOperationsonsets,characteristicfunctions.Maps,injectivity,surjectivity.Directandinverseimageofaset.Integernumbers,finitesets,countability.1.2NumbersandusualstructuresComposi
2、tionlaws;groups,rings,fields.Equivalencerelations,quotientstructures.Realnumbers,complexnumbers,complexexponential.Applicationtoplanegeometry.Polynomials,relationsbetweentherootsandthecoefficients.Elementaryarithmetics(inZ/nZ).1.3Finitedimensionalvectorspaces(*)F
3、reefamilies,generatingfamilies,bases,dimension.Determinantofnvectors;characterizationofbases.Matrices,operationsonmatrices.Determinantofasquarematrix;expansionwithrespecttoalineortoacolumn;rank,cofactors.Linearmaps,matrixassociatedtoalinearmap.Endomorphisms,trace
4、,determinant,rank.Linearsystemsofequations.1.4ReductionofendomorphismsStablesubspaces.Eigenvalues,eigenvectorsofanendomorphismorasquarematrix;similarmatrices;geometricalinterpretation.Characteristicpolynomial,Cayley-Hamiltontheorem.Reductionofendomorphismsinfinit
5、edimension;diagonalizableendomorphismsandmatrices.1.5Euclideanspaces,EuclideangeometryScalarproduct;Cauchy-Schwarzinequality;normsandassociateddistances.Euclideanspacesoffinitedimension,orthonormalbases;orthogonalprojections.OrthogonalgroupO(E);orthogonalsymmetri
6、es.Orthogonalmatrices;diagonalizationofsymmetricrealmatrices.PropertiesoforthogonalendomorphismsofR²andR³.kk(*)InseveralcountrieslinearalgebraisstudiedonlyinRorC;thecandidatesfromthesecountriesarestronglyadvisedtogetfamiliarwiththeformalismofabstractvectorspaces.
7、2-ANALYSISANDDIFFERENTIALGEOMETRY2.1TopologyinfinitedimensionalnormedvectorspacesOpenandclosedsets,accumulationpoints,interiorpoints.Convergentsequencesinnormedvectorspaces;continuousmappings.Compactspaces,imagesofcompactsetsbycontinuousmappings,existenceofextrem
8、a.Equivalenceofnorms.2.2RealorcomplexvaluedfunctionsdefinedonanintervalkDerivativeatapoint,functionsofclassC.Meanvaluetheorem,Taylor'sformula.Primitiveofcontin
