Thisbookalmstointroducethebeginningorworkingphysicisttoawiderangeofaualytictoolswhichhavetheiror/ginindifferentialgeometryandwhichhaverecentlyfoundincreasinguseintheoreticalphysics.Itisnotuncom-montodayforaphysicistsmathematicaleducationtoignoreallbutthesim-plestgeometricalideas，despitethefactthatyoungphysicistsareencouragedtodevelopmentalpicturesandintuitionappropriatetophysicalphenomena.Thiscuriousneglectofpicturesofonesmathematicaltoolsmaybeseenastheoutcomeofagradualevolutionovermanycenturies.Geometrywascertainlyextremelyimportanttoancientandmedievalnaturalphilosophers;itwasingeometricaltermsthatPtolemy，Copernicus，Kepler，andGalileoallexpressedtheirthinking.ButwhenDescartesintroducedcoordinatesintoEuclideangeometry，heshowedthatthestudyofgeometrycouldberegardedasanappli.cationofalgrebra.Sincethen，the/mportanceofthestudyofgeometryintheeducationofscientistshassteadily