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1. f: R x R -> R is differentiable such that df/dx=df/dy, and for all x in R, f(x,0)>0. Show that for all x,y in R, f(x,y)>0.
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2. f: [0,1] -> R is C^1 and f(0)=0. Show that sup(|f(x)|,x \in [0,1]) <= sqrt(int(f '^{2}(t)dt,t=0..1)).
Solution

3. Let f: R^n -> R be a function with continuous partial derivatives such that for all 1 <= i <= n, for all x in R, |df/dx_i(x)| <= K. Show that |f(x)-f(y)| <= sqrt(n)K.||x-y||_n.
Solution

4. Let f: R -> R be a continuously differentiable function such that f(0)=1, 0>f '(0) and for all x in (0,1], 0 <= f(x) < 1. Show that: limit n int(f^n(x)dx,x=0..n)=-1/f '(0). Solution

5. Assuming X_2 is a differentiable manifold with finite dimension and delta: X_1 -> X_2 is a continuous function from a differentiable manifold X_1 into X_2 such that f o phi is differentiable for every locally defined differentiable real-valued function f, prove that delta is differentiable.
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6. M_1,M_2 are two differentiable manifolds. Let f be the flow generated by a smooth vector field X on M_2, and let g: M_1 -> M_2 be a diffeomorphism. Show that the flow generated by g*[X] (the pullback vector field on M_1) is given by h(x,t)=g^(-1)f(g(x),t).
Solution