2. Let U be an open subset of R and let A be a compact subset of U. Suppose that f: U R is a iction of class C() aud...
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.
Let U be an open subset of R". Let f:UCR"-R be differentiable at a E U. In this exercise you will prove that if ▽f(a) 0, then at the point a, the function f increases fastest in the direction of V f(a), and the maximum rate of increase is Vf(a)l (a) Prove that for each unit vector u e R" (b) Prove that if ▽/(a)メ0, and u = ▽f(a)/IV/(a) 11, then Let U be an open subset of R". Let...
Theorem 2. Let E be an open subset of R² and suppose that fe C'(E). Let y(t) be a periodic solution of (1) of period T. Then the derivative of the Poincaré map P(8) along a straight line normal to r = {x E R x = y(t) - (0),O SE ST} at x = 0 is given by T P(0) = exp V. f(y(t)) dt. 4. Show that the system • = -y + (1 – 22 - y2)2...
1 Let f: R R be a continuously differentiable map satisfying ilf(x)-FG) ll 리1x-vil, f Rn. Then fis onto 2. f(RT) is a closed subset of R'" 3, f(R") is an open subset of RT 4. f(0)0 or all x, y E 5) S= (xe(-1,4] Sin(x) > 0). Let of the following is true? I. inf (S).< 0 2. sup (S) does not exist Which . sup (S) π ,' inf (S) = π/2 1 Let f: R R be...
B is a connected ball of finite radius 2, Let f : U → Rm be Ci and let B be a compact connected subset of U Show that there exists a constant M such that for all a, y e B. (Hint: use the mean value theorem). Find an example which shows that the assumption that B was compact is essential 2, Let f : U → Rm be Ci and let B be a compact connected subset of...
*4, Let U be an open subset of R" and f:U-R" a function whose component functions have continuous partial derivatives. We say that f is an immersion if Dsf is injective for all v in U and a submersion if Dof is surjective for allv in U. (a) Suppose that f:U-R" is an immersion. Prove that, for each v in U, we can find an open set V of U containing v, an open set W of R" containing f...
1. Assume that S is an open subset of R", and that f, g: S R" are functions of class C in S. Prove that := f.g : S R is of class C, and that - D g) (Df)'g + (Dg)t (8) where T denotes "transpose." 1. Assume that S is an open subset of R", and that f, g: S R" are functions of class C in S. Prove that := f.g : S R is of class...
Let U ⊆ R^n be open (not necessarily bounded), let f, g : U → R be continuous, and suppose that |f(x)| ≤ g(x) for all x ∈ U. Show that if exists, then so does . We were unable to transcribe this imageWe were unable to transcribe this image
1) Show that if U is a non-empty open subset of the real numbers then m(U) > O. 2) Give an example of an unbounded open set with finite measure. Justify your answer, 3) If a is a single point on the number line show that m ( a ) = O. 4) Prove that if K is compact and U is open with K U then m(K) m(U). 5) show that the Cantor set C is compact and m(C)...