Fizika-matemati fakulteti

(Wⁱ(V_j)=δ_jⁱ)

wⁱ_1*******Wⁱ_k=V_j1******V_jk 1 ≤ jp, iq ≤ n,

asosini tashkil etadi T_l^k (V )

dimT^k_l(V ) =n^k+l

Chunki T1(V ) = V= ∗ ∗ (Ya'ni, har bir vektor v ∈ V 1-contravariant tensor bir emas) V vaT1(V ) = V∗(har bir kovektor 1-kovariant tenzor), bizda

wⁱ_1*******Wⁱ_k=V_j1******V_jk 1 ≤ jp, iq ≤ n,

a (k, l)-tensor bo'ladi.

2.2 Cotangent to'plam Tensor to'plamlar

Funksiyaning differensialini ilgari aniqladik f ∈ C∞(p) p da chiziqli xarita

dfp : TpM → R, dfpv = vf, v ∈ TpM

Shuning dfp ∈ TpM∗ (= TpM hamda dual). TpM∗biz qo'ng'iroq

M cotangent oraliq p da Agar

(U, x), x = (x1, . . . , xn), p va District da grafikdir(∂1)p, . . . ,(∂n)p TpM ning asosini tashkil etadi koordinata vektorlari, so'ngra differensiallar dxip, i = 1, . . . , n, vazifalari x p da i ning dual asosini tashkil TpM∗

. Demak, f ∈ C∞(p) funksiyaning differensiali(p da) deb yozish mumkin dfp = (∂i)pf dxi p .

M ning kotangent to'plamini M ning barcha kotangent fazolarining disjoint birlashmasi deb belgilaymiz

TM∗ = G p∈M TpM


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(TM deb shunga o'xshash belgilangan) tabiiy C∞-tuzilishi bilan jihozlangan. Bundan tashqari, ruxsat π :∗ → M,∗ ∋ 7→ p ω TpM ∈ M canonical bu loyiha bo'lishi TM.

TM∗bo'limlarini chaqiramiz, ya'ni mappings ω: M → TM∗, = π ◦ ω id bilan, M sohalarda covector yoki (differensial) 1-shakllari. Hoshi. bizdenote T tomonidan1(M) (yoki T10(M), T∗(M), T0,1(M)) M barcha silliq covector sohalarda majmui.funksiyaning differensiali f ∈ C∞(M) (silliq) kovektor maydon
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