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PROBLEM SET VIII DUE FRIDAY, APRIL Exercise . Use the Poincaré lemma to show that there exists a smooth function g : R . R such that ′ (sin x + x exp(g(x)) − 1)g (x) + g(x) cos x + 2x exp(g(x)) = 0. 2 Deﬁnition. For any real numbers 0 ≤ r ≤ s, let T2 (r, s) ⊂ R3 denote the torus { } T2 (r, s) := (x, y, z) ∈ R3 (x2 + y2 + z2 + s2 − r2 )2 = 4s2 (x2 + y2 ) . ∪ Let us write ST(r, s) for the union ρ∈[0,r] T2 ( ρ, s). Exercise . What is the volume of ST(r, s)? Exercise . What is the surface area of T2 (r, s)? Exercise⋆ . Consider the following vector ﬁeld on R3 − {(0, 0, 0)}: 1 φ(x, y, z) = 2 (−y, x, (z − 1/4)(z + 1/4) − x2 − y2 ). x + y2 + z2 (a) Make a good faith attempt to draw this vector ﬁeld (or some ﬂow lines). (b) Is this vector ﬁeld a gradient vector ﬁeld?