Integration & Stoke's Theorem

Question 1

SMWB

Answer 1

For every p in M there is a diffeomorphism from V (open subset of H^n) to U (open nbhd of p)

Question 2

parametrisation

Answer 2

diffeomorphism φ

Question 3

T_pM

Answer 3

span of γ’(0) over all smooth curves with γ(0)=p and γ(t) in Μ

Question 4

Heine-Borel

Answer 4

a subset of a finite dimensional normed vector space is compact iff closed and bounded

Question 5

support of f

Answer 5

closure in S (domain of f) of the set {p in S: f(p) =/ 0}

Question 6

multiple integral

Answer 6

inductively via f~

Question 6

Jacobian

Answer 6

J_φ_p= det(Dφ_p)

Question 7

Change of variables formula for multiply integrals

Answer 7

f o φ times absolute value of Jacobian

Question 8

orientation preserving

Answer 8

always positive Jacobian

Question 9

Change of variables for differential forms

Answer 9

φ orientation-preserving diffeomorphism and a with compact support

Question 10

orientation form on SMWB

Answer 10

never vanishes

Question 11

orientable SMWB

Answer 11

orientation form exists

Question 12

orientation on M

Answer 12

equivalence class on orientation forms (equivalent for all p)

Question 13

oriented SMWB

Answer 13

M together with a choice of orientation

Question 14

local diffeomorphism of oriented SMWBs is orientation preserving

Answer 14

for an orientation form ω defining orientation on M, φ*ω defines chosen orientation on N

Question 15

integration map

Answer 15

linear
φ oriented parametrisation, α compact support, supp(α) in U, then φ*α defined

Question 16

partition of unity on S subordinate to open cover U_i

Answer 16

indexed family ρ_i such that
each ρ_i nonnegative smooth function
supp(ρ_i) contained in U_i
each p in S has a nbhd U st its intersection with supp(ρ_i) is nonempty only for finitely many I
sum of ρ_i at p equals 1

Question 17

boundary-less case of stoke’s theorem

Answer 17

M oriented n-manifold and β compact n-1, then integral over M of dβ is 0