+ h ▁lies ▁entirely ▁within ▁U . ▁ ▁If ▁F ▁is ▁C k ▁then ▁ ▁where ▁the ▁remainder ▁term ▁is ▁given ▁by ▁▁ ▁Comm ut ativity ▁of ▁direction al ▁derivatives . ▁ ▁If ▁F ▁is ▁C k , ▁then ▁ ▁for ▁every ▁permut ation ▁ σ ▁of ▁{ 1 , 2 ,..., k }. ▁ ▁The ▁proofs ▁of ▁many ▁of ▁these ▁properties ▁rely ▁fund ament ally ▁on ▁the ▁fact ▁that ▁it ▁is ▁possible ▁to ▁define ▁the ▁Riemann ▁integral ▁of ▁continuous ▁curves ▁in ▁a ▁Fr éch et ▁space . ▁ ▁Sm ooth ▁ma ppings ▁Sur pr ising ly , ▁a ▁mapping ▁between ▁open ▁subset ▁of ▁Fr éch et ▁spaces ▁is ▁smooth ▁( inf initely ▁often ▁differenti able ) ▁if ▁it ▁maps ▁smooth ▁curves ▁to ▁smooth ▁curves ; ▁see ▁Con ven ient ▁analysis . ▁Moreover , ▁smooth ▁curves ▁in ▁spaces ▁of ▁smooth ▁functions ▁are ▁just ▁smooth ▁functions ▁of ▁one ▁variable ▁more . ▁ ▁Con sequ ences ▁in ▁differential ▁geometry ▁The ▁existence ▁of ▁a ▁chain ▁rule ▁allows ▁for ▁the ▁definition ▁of ▁a ▁manifold ▁mode led ▁on ▁a ▁Fr è ch et ▁space : ▁a ▁Fr éch et ▁manifold . ▁ ▁Furthermore , ▁the ▁linear ity ▁of ▁the ▁derivative ▁implies ▁that ▁there ▁is ▁an ▁analog ▁of ▁the ▁tang ent ▁bundle ▁for ▁Fr éch et ▁manif olds . ▁ ▁T ame ▁Fr éch et ▁spaces ▁Fre qu ently ▁the ▁Fr éch et ▁spaces ▁that ▁arise ▁in ▁practical ▁applications ▁of ▁the ▁derivative ▁enjoy ▁an ▁additional ▁property : ▁they ▁are ▁t ame . ▁ ▁R ough ly ▁speaking , ▁a