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 F-(()e KIf(r, y) 0 and that Df does not vatish on E. Investigate whether Dis a Jordan region. annc
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.
*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...
Please prove by setting up the theorem below (Chain Rule)
v:RR is continuously differentiable. Define the Suppose that the function function g : R2R by 8(s, t)(s2t, s) for (s, t in R2. Find ag/as(s, t) and ag/at(s, t) Theorem 15.34 The Chain Rule Let O be an open subset of R and suppose that the mapping F:OR is continuously differentiable. Suppose also thatU is an open subset of Rm and that the functiong:u-R is continuously differentiable. Finally, suppose that...
2. Let f:R + R and g: R + R be functions both continuous at a point ceR. (a) Using the e-8 definition of continuity, prove that the function f g defined by (f.g)(x) = f(x) g(x) is continuous at c. (b) Using the characterization of continuity by sequences and related theorems, prove that the function fºg defined by (f.g)(x) = f(x) · g(x) is continuous at c. (Hint for (a): try to use the same trick we used to...
(8) Let E C R" and G C R" be open. Suppose that f E G and g G R', so that h = go f : E → R. Prove that if f is differentiable at a point x E E, and if g is differentiable at f (x) E G, then the partial derivatives Dihj(x) exist, for all and j - ...., and 7m に! (The subscripts hi. g. fk denote the coordinates of the functions h, g....
3-2. Prove Theorem 3.2. Theorem 3.2 Let I S R be an open interval, xe I, and let f. 8:1\{x} → R be functions. If there is a number 8 > 0 so that f and g are equal on the subset 12 € 7\(x): 13-X1 < 8 of I\(x), then f converges at x iff g converges at x and in this case the equality lim f(x) = lim g(z) holds.
How would I prove 1.4. What is this called? I can't
find these properties in my textbook. What is the name of this
stuff?
provided integrals on the right exist. If g is a non-increasing function, we have the | again provided the integrals on the right exist. Vte(a,b], l(t)< h(t) → l(t) do(t)2/ h(t) dg(t), a. Throghout these noteill we h oloring dofinition Definition 1.3. Let a, b E R with a < b and let k Zco. We...
(8) Let E c R" and G C Rm be open. Suppose that f E -G and g:GR', so that h -gof:E R'. Prove that if f is differentiable at a point x E E and if g is differentiable at f(x) є G, then the partial derivatives Dh,(x) exist, for all , SO , . . . , n, and and J-: に1 The subscripts hi, 9i, k denote the coordinates of the functions h, g, f relative to...