Thistextisanelementaryintroductiontodifferentialgeometry.Althoughitwaswrittenforagraduate-levelaudience,theonlyrequisiteisasolidback-groundincalculus,linearalgebra,andbasicpoint-settopology.　　Thefirstchaptercoversthefundamentalsofdifferentiablemanifoldsthatarethebreadandbutterofdifferentialgeometry.Alltheusualtopicsarecovered,culnunatinginStokes'theoremtogetherwithsomeapplications.Thestudents'firstcontactwiththesubjectcanbeoverwhelmingbecauseofthewealthofabstractdefinitionsinvolved,soexampleshavebeenstressedthroughout.Oneconcept,forinstance,thatstudentsoftenfindconfusingisthedefinitionoftangentvectors.Theyarefirsttoldthatthesearederivationsoncertainequiv-alenceclassesoffunctions,butlaterthatthetangentspaceofRlis"thesame"asRn.WehavetriedtokeepthesespacesseparateandtocarefullyexplainhowavectorspaceEiscanonicallyisomorphictoitstangentspaceatapoint.Thissubtledistinctionbecomesessentialwhenlaterdiscussingtheverticalbundleofagivenvectorbundle.