Differential Forms

Question 1

alternating (multilinear form)

Answer 1

α(v_1,…,v_k)=0 whenever v_i=v_j

Question 2

degree of α

Answer 2

k

Question 3

σ*α

Answer 3

α(v_σ(1),…,v_σ(k))

Question 4

alt(α)

Answer 4

Σ (over σ) sgn(σ) σ*α

Question 4

α_1…α_k

Answer 4

α_1(v_1)…α_k (v_k)

Question 5

α_1 ^…^ α_k

Answer 5

alt(α_1…α_k)

Question 6

Smooth differential k-form

Answer 6

smooth function from U to Alt^k (RR^n) sending p to α_p

Question 7

φ*α

Answer 7

α(φ(v_1), … , φ(v_k))

Question 8

pullback

Answer 8

(Dφ_p)* α_φ(p)

Question 9

exterior derivative

Answer 9

(dα)_p = 1/k! alt((Dα)_p^)

Question 10

αβ(v_1,…,v_k+l)

Answer 10

α(v_1,…,v_k) β(v_k+1,…,v_k+l)

Question 11

Poincare’s Lemma

Answer 11

closed α in Alt^k (RR^n) implies exact

Question 12

α^β

Answer 12

1/k!l! alt(αβ)

Question 13

wedge product

Answer 13

(α^β)_p = α_p ^ β_p

Question 14

differential k-form on submanifold of dimension m

Answer 14

function α from M to A_m ^k (RR^s) st
α_p is in Alt^k (T_pM) for all p
for all p there is open nhbd U in RR^s st for all q in intersection of U with M
α_q= β_q restricted to T_qM^k

Question 15

exterior derivative

Answer 15

whenever β is a local extension of α on intersection of M with U then dα=ι*dβ where ι is the inclusion map