Final Exam Solution

# If 1 and 2 are both k forms then f 1 2 f 1

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Unformatted text preview: . 3. Also let σ1 , • For each k there is a zero k -form with the property 0 + ω = ω and 0 ∧ η = 0 for all ω and η . • If ω1 and ω2 are both k -forms then (f ω1 + ω2 ) ∧ η = f (ω1 ∧ η ) + (ω2 ∧ η ) • ω ∧ η = (−1)kℓ (η ∧ ω ) • ω 1 ∧ (ω 2 ∧ ω 3 ) = (ω 1 ∧ ω 2 ) ∧ ω 3 • ω ∧ (f η ) = (f ω ) ∧ η = f (ω ∧ η ) • dx ∧ dy = dx dy , dy ∧ dz = dy dz , dz ∧ dx = dz dx • dx ∧ dx = dy ∧ dy = dz ∧ dz = 0 • dx ∧ (dy ∧ dz ) = dx dy dz = (dx ∧ dy ) ∧ dz • f ∧ ω = fω • df = ∂f dx ∂x + ∂f dy ∂y + ∂f dz . ∂z • If ω1 and ω2 are k -forms then d (ω1 + ω2 ) = dω1 + dω2. • d (ω ∧ η ) = (dω ∧ η ) + (−1)k (ω ∧ dη ) • d (dx) = d (dy ) = d (dz ) = 0. Math 3010 Final Exam - Page 3 of 14 Dec. 8, 2012 1. Improper integrals (a) (6 points) Let D = {(x, y ) |x ∈ [a, b] , ϕ1 (x) y ϕ2 (x)} where ϕi are continuous. Let f : R2 → R be continuous in the interior of D but possibly discontinuous on points on ∂D . Deﬁne the following improper integrals: ϕ2 (x) and f (x, y ) dy dx. ϕ1 ( x ) D ϕ2 ( x ) b f (x, y ) dy f dA , ϕ1 (x) a Solution: Let Dη,δ = {(x, y ) |x ∈ [a + η, b − η ] , y ∈ [ϕ1 (x) + δ, ϕ2 (x) − δ ]}. Let η, δ > 0. b−η f dA = D f dA = lim (η,δ)→(0,0) Dη,δ f (x, y ) dy dx. (η,δ)→(0,0) a+η ϕ2 ( x ) ϕ1 (x)+δ ϕ2 (x)−δ f (x, y ) dy = lim δ→0 ϕ1 (x)+δ ϕ1 (x) b ϕ2 (x)−δ lim ϕ2 ( x ) b−η f (x, y ) dy. ϕ2 (x) f (x, y ) dy dx. =...
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## This note was uploaded on 01/19/2014 for the course MATH 3010 taught by Professor Magpantay during the Winter '13 term at York University.

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