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GEOM2, F2007. EXERCISES 8b. Exercises for Friday Mar 30 1 Let ξ denote the covector field 2xy dx + (x2 + y) dy on R2 . Verify that ξ is closed, and find f ∈ C ∞ (R2 ) such that df = ξ. P ∂ai = 2 A covector field ξ = i ai dxi on Rn is said to be closed if ∂x j pairs of indices i, j. Consider the covector field ∂aj ∂xi for all (y 2 + 2xz + 2x)dx + (z 2 + 2xy + 2y)dy + (x2 + 2yz + 2z)dz on R3 . Verify that it is closed, and find f ∈ C ∞ (R3 ) such that df = ξ. 3 Let M be an abstract manifold, and let f ∈ C ∞ (M ). Assume that df = 0. Prove that each level set {p ∈ M | f (p) = c} is both closed and open. Conclude that f is constant on each component of M , and show that the dimension of the kernel of d in C ∞ (M ) is the number of components (which is either a finite number or infinite). 4 Let M = R2 \ {(x, 0) | x ≤ 0}, and let (r, ϕ) denote the polar coordinates of (x, y), that is, (x, y) = (r cos ϕ, r sin ϕ). Here ϕ ∈] − π, π[. a) Determine dx and dy in terms of r, ϕ, dr and dϕ. b) Determine dr and dϕ in terms of x, y, dx and dy. c) Show that the covector field ξ= x −y dx + x2 + y 2 x2 + y 2 is closed on R2 \ {(0, 0)}, and that it is exact on M but not on R2 \ {(0, 0)}. 5 Let ξ be a covector field on an abstract manifold M . Prove that ξ is smooth if ξ(Y ) ∈ C ∞ (M ) for all Y ∈ X(M ) (thus improving the ‘if’ of Lemma 8.2.1). 6 (if there is time) Prove Lemma 8.3.