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18.755 problems due 9/21/15 in class 1. Suppose f ∈ C ∞ (R), and f (0) = 0. Prove that there is a function g ∈ C ∞ (R) with the property that g(x) = f (x)/x (x 6= 0). What is g(0) (in terms of f )? Possible hint: compute Z 1 f ′ (tx)dt. 0 2. Write m0 = {f ∈ C ∞ (R) | f (0) = 0}. Prove that m20 = {h ∈ C ∞ (R) | h(0) = h′ (0) = 0}. For the next problems, recall that a path in a manifold is a smooth map γ: (a, b) → M. The derivative or velocity vector at t ∈ (a, b) is γ ′ (t) ∈ Tγ(t) M, a (real) tangent vector. Recall also that a (real) vector field X on M assigns to each m ∈ M a tangent vector Xm ∈ Tm M. We say that γ is an integral curve of X if γ ′ (t) = Xγ(t) (all t ∈ (a, b)). Suppose U ⊂ M is open, and φ: U → Rn , φ(m) = (x1 (m), . . . , xn (m)) is a local coordinate system (as in the definition of manifold). Then the partial derivatives ∂/∂x1 , . . . , ∂/∂xn are a basis of Tm (M ) for every m ∈ U . The restriction to U of a (smooth) vector field is specified by m smooth functions a1 , . . . , am in C ∞ (U ) so that ∂ ∂ + · · · + an (m) . ∂x1 ∂xn A path in U is specified by n smooth functions γ1 , . . . , γn , Xm = a1 (m) (γ1 (t), . . . , γn (t)) = φ(γ(t)) ∈ φ(U ) (t ∈ (a, b)). To say that γ is an integral curve for X amounts to the system of differential equations dγp (t) = ap (γ1 (t), . . . , γn (t)) (p = 1, . . . , n). dt The basic fact about integral curves, discussed in class Wednesday 9/16 (I hope!) is 1 2 Proposition. Suppose X is a (smooth) vector field on M . For every (m0 , t0 ) ∈ M × R, there is an open interval (a, b) containing t0 and an integral curve γm0 ,t0 of X satisfying the initial condition γ(t0 ) = m0 . If γ1 and γ2 are two such integral curves defined on intervals (a1 , b1 ) and (a2 , b2 ), then γ1 = γ2 on (a1 , b1 ) ∩ (a2 , b2 ). Consequently there is a unique largest interval (a(m0 , t0 ), b(m0 , t0 )) on which the integral curve exists; here −∞ ≤ a(m0 , t0 ) < t0 < b(m0 , t0 ) ≤ +∞. 3. Let M be the real line; I’ll write x for a variable point of M . Consider the three vector fields Xx(0) = d , dx Xx(1) = x d , dx Xx(2) = x2 d . dx For every real number t0 and every x0 ∈ M (also a real number) calculate the (p) maximal integral curves γx0 ,t0 for each of the three vector fields X (p) (p = 0, 1, 2). Hint: this is a problem about ordinary differential equations. One of the differential equations you need to solve is dγ = γ 2 (t). dt 4. Let M = R2 be the real plane; I’ll write (x1 , x2 ) for a variable point of M . Consider the three vector fields Xx(0) = x1 1 ,x2 ∂ ∂ + x2 , ∂x1 ∂x2 Xx(1) = x1 1 ,x2 ∂ ∂ − x2 , ∂x1 ∂x2 Xx(2) = x1 1 ,x2 ∂ ∂ − x2 . ∂x2 ∂x1 For every real number t0 and every (x1,0 , x2,0 ) ∈ M calculate the maximal integral curves (p) (p) (p) γ(x1,0 ,x2,0 ),t0 = (γ1,(x1,0 ,x2,0 ),t0 , γ2,(x1,0 ,x2,0 ),t0 ) for each of the three vector fields X (p) .