c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 ω • space Dual 5 Chapter as h ulsaei etrsaeudrteoeain fvco ad- vector of operations by the defined under multiplication space scalar vector and a dition is space dual The P • mappings linear vectors cotangent • : rdcsanumber, a produces h T ewl rt h cinof action the write will We iert ftemapping the of Linearity Example: space vector any for defined be can space dual A ω The ie ucino manifold a on function a Given of elements The P | → M v ω P ( ulspace dual u i P . a R + 1 . ω av 1 ucin ierfntoas etc. functionals, linear vector Dual vector row functions kets vectors column Vector V P + = ) → a | etc. 2 ψ ω T R i T ω 2 P v ∗ P (or ∗ ( P M : u M ( v P f P V + ) of ) ω 7→ ω r called are ∈ → T means on aω P ∀ a R 16 1 u C M ω v ( P v P 1 if v , ∀ P ( stesaeo iermappings linear of space the is v v ∈ f V ) P P P bras , T : + ) ∈ sacmlxvco space). vector complex a is ∈ ulvectors dual P → M T M T h P a P φ 2 M as | ω M 2 ω ( R . and v ( P Thus , v V ) P vr etrat vector every . rsometimes or ) stesaeof space the as a , ∈ covectors f R ensa defines . 2 (5.1) (5.2) 2 2 , c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 ω d the So words, other In vectors. basis the of one every on gradient d covector h d the So get constructed have function smooth a is d Thus aino hs covectors, these of nation ∂x λ ial,gvnaycovector any given Finally, Proposition: Since h ieetasd differentials The Proof: ∂ d x i aihso l etr,snethe since vectors, all on vanishes x i i f P λ span x d ∈ v i ∂x x of , f P d ∂ r ieryindependent. linearly are T i ∂x osdrachart a Consider f j . ∂ slna,s sd is so linear, is f P ∗ T ( ie yd by given hnltigti c nacodnt ai etr we vector, basis coordinate a on act this letting Then j . M v P ∗ P P M P ω n . + = T , oetr in covectors 0 = = = aw P ∗ so x M ∂x x ω ω i P ∂ T i : ⇒ ⇒ ⇒ ( = ) j P f salso is ∗ r oetr,a earayko.S we So know. already we as covectors, are → M = ω ( ∂x ∂x M ω ω ω v ∂ ∂ P = P v , f i i ϕ j j d δ v ( P = ) is j x i x P ω ihcodnt functions coordinate with ∂x ( R i T 0 = i P P f n ∂ i n = ) , ω + d ∀ P + ) -dimensional. ∗ -dimensional. . j − − x v v M aw ∂x hntedffrnil d differentials the then i P P osdrtecovector the consider ∂ . ω ω P ∂x w , aw ( fti aihs tms vanish must it vanishes, this If j P ..ω i.e. . f ∂ 0 = )( etcnie iercombi- linear a consider Next aldthe called ) j P P ∂x ∂x P f ∂ ∂ ∂x ( ∈ x ) f ∂ 0 = i i i (5.3) ) T j j ◦ P P P 0 = ϕ M P δ d − j i x omabss Thus basis. a form 1 a , 0 = i .
ϕ differential ∈ ∂x ( ∀ P ∂ R j ) j . = x x i P λ δ i . j i Each satisfy = . (5.5) (5.6) (5.4) ω 17 or x − 2 i c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 rmwihi olw that follows it which from chart new a y In transformation. coordinate ϕ a i.e. chart, overlapping gradient. name the justifies This ie basis a given for particular in so with bases. arbitrary to generalized be can formulae These for covec- of basis a on transformations coordinate of tors. result the is This as 18 i 0 ( hr h oriae are coordinates the where x ti tagtowr ocluaeteeeto wthn oanother to switching of effect the calculate to straightforward is It ie vector a Given Since covector any seen, just have we as Also, ecnmk hneo ae yalna transformation, linear a by bases of change a make can We ie 1-form a Given )w a s q 58 owietegain of gradient the write to (5.8) Eq. use can we )) A ∂y o-iglrmti,s that so matrix, non-singular a ∂ ω i a 7→ P ω { ∂x ω ∂ e i ω = a i ≡ ob h ulbssto basis dual the be to 0 } a , ω λ = (d P v ecndfieisda basis dual its define can we i λ ω d ti o ennflt akaotisda,but dual, its about talk to meaningful not is it , ∂y = f A ∂ x ecnwiei nbt bases, both in it write can we d = ) i i λ b a steda ai in basis dual the is i d ω a d = y ω b P , f i e , a λ = = where a 0 f = eget we y ( = i λ adtetasto ucin r thus are functions transition the (and ∂x ∂x ∂x ∂y ∂y a 0 ∂ A ω i j j i i 0 − a ω 1 = P ) P P i a a b ω = λ d = λ 7→ { 0 b x a 0 a d ω ∂x A ( . j y ∂ e e ω T i b a a 0 ∂x b 0 j ∂f } ω = ) P hpe .Da space Dual 5. Chapter ∂x ( = ∈ a ems have must we ∂ M i P { , T i ω δ A ϕ P ∗ b a a to ( − P M } P . 1 y ) by , ) { i a b a ewritten be can d e as x ω b i , a } ( norder in , e b = ) (5.12) (5.10) (5.11) (5.8) (5.7) (5.9) δ b a . c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 hs rnfriglike transforming those • that follows it and iial,if Similarly, uniiswihtasomlike transform which Quantities v v savco,w a write can we vector, a is = v a v = a e v A a a b a = r called are v b v . 0 a e a 0 = v contravariant λ 0 a a ( A r called are − 1 ) a b e b , covariant . while , (5.13) 19 2