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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 ea...
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...
Under is for reference (Mean Value Theorem): Suppose that f: R6 + R is a function...
Under is for reference (Mean Value Theorem):
Suppose that f: R6 + R is a function with the following two properties: flo) = 0, and at at any point Te R6 and any increment h, || DFOD | E || ||. Show that f(B1)) (-1,1). Comment. You should use the Mean Value Theorem at some point in this problem. An interpretation with more jargon is that if the operator norm of Df is at most 1 at all points, then...
(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...
(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...
4. Prove the following statement: Consider the ODE x = f(x) with x : J C R → Rn and f : Rn → Rn. ...
4. Prove the following statement: Consider the ODE x = f(x) with x : J C R → Rn and f : Rn → Rn. If a continuously differentiable real-valued function V = V(x) exists such that (a) V is defined on Bs(0) {x E Rn : Irl < δ} (b) V(x) 0 for x E Bs(0) 1 fo) (c) V 0) 1 (o then the origin is unstable. (x) >0 for rE Bs
4. Prove the following statement: Consider...
Monotone mappings. A function u : Rn Rn is called monotone if for all x, y є dom v, Note that monotone' as defined here is not the same as the definition given in 83.6.1. Both definitions are wid...
Monotone mappings. A function u : Rn Rn is called monotone if for all x, y є dom v, Note that monotone' as defined here is not the same as the definition given in 83.6.1. Both definitions are widely used.) Suppose f R"- R is a differentiable convex function. Show that its gradient ▽f is monotone. Is the converse true. i.e., 1s every monotone mapping the gradient of a convex function?
Monotone mappings. A function u : Rn Rn is...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and conti...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
1. Let U с Rn be open, f : U-> Rm be a function, a є U and 0 exists. Show that DAwf(a) exists for every 0メλ R, and DAwf(a) Rn such that Duf(a) λDuf(a). 3 marks 1. Let U с Rn be open, f : U-&...
1. Let U с Rn be open, f : U-> Rm be a function, a є U and 0 exists. Show that DAwf(a) exists for every 0メλ R, and DAwf(a) Rn such that Duf(a) λDuf(a). 3 marks
1. Let U с Rn be open, f : U-> Rm be a function, a є U and 0 exists. Show that DAwf(a) exists for every 0メλ R, and DAwf(a) Rn such that Duf(a) λDuf(a). 3 marks
Q5 (a) Provide the definition of the derivative of a map F: ViV2 where (V, l1) a are normed vector spaces (possibly...
Q5 (a) Provide the definition of the derivative of a map F: ViV2 where (V, l1) a are normed vector spaces (possibly infinite dimensional) (b) Let C((0, 1) be the space of continuous real valued functions on [0, 1] endowed with the supremum norm. Define F:C ((0, 1]) C([0, 1]) by F(() Jo f()dt, e for all f E C(0, 1). Show directly from the definition that the derivative of F is differentiable on the entire domain. (c) For the...
V:RR is continuously differentiable. Define the Suppose that the function function g : R2R by 8(s...
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...
please show all work, even trivial steps. Here are definitions if needed. do not write in...
please show all work, even
trivial steps. Here are definitions if needed. do not write in
script thank you!
4. Letf: R2 → R2, by f(x,y) = (x-ey,xy). a. Find Df (2,0). b. Find DF-1(f (2,0)) Inverse Function Theorem: Suppose that f:R" → R" is continuously differentiable in an open set containing a and det(Df(a)) = 0, then there is an open set, V, containing a and an open set, W, containing f(a) such that f:V W has a continuous...
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...
Under is for reference (Mean Value Theorem):
Suppose that f: R6 + R is a function with the following two properties: flo) = 0, and at at any point Te R6 and any increment h, || DFOD | E || ||. Show that f(B1)) (-1,1). Comment. You should use the Mean Value Theorem at some point in this problem. An interpretation with more jargon is that if the operator norm of Df is at most 1 at all points, then...
(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...
4. Prove the following statement: Consider the ODE x = f(x) with x : J C R → Rn and f : Rn → Rn. If a continuously differentiable real-valued function V = V(x) exists such that (a) V is defined on Bs(0) {x E Rn : Irl < δ} (b) V(x) 0 for x E Bs(0) 1 fo) (c) V 0) 1 (o then the origin is unstable. (x) >0 for rE Bs
4. Prove the following statement: Consider...
Monotone mappings. A function u : Rn Rn is called monotone if for all x, y є dom v, Note that monotone' as defined here is not the same as the definition given in 83.6.1. Both definitions are widely used.) Suppose f R"- R is a differentiable convex function. Show that its gradient ▽f is monotone. Is the converse true. i.e., 1s every monotone mapping the gradient of a convex function?
Monotone mappings. A function u : Rn Rn is...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
1. Let U с Rn be open, f : U-> Rm be a function, a є U and 0 exists. Show that DAwf(a) exists for every 0メλ R, and DAwf(a) Rn such that Duf(a) λDuf(a). 3 marks
1. Let U с Rn be open, f : U-> Rm be a function, a є U and 0 exists. Show that DAwf(a) exists for every 0メλ R, and DAwf(a) Rn such that Duf(a) λDuf(a). 3 marks
Q5 (a) Provide the definition of the derivative of a map F: ViV2 where (V, l1) a are normed vector spaces (possibly infinite dimensional) (b) Let C((0, 1) be the space of continuous real valued functions on [0, 1] endowed with the supremum norm. Define F:C ((0, 1]) C([0, 1]) by F(() Jo f()dt, e for all f E C(0, 1). Show directly from the definition that the derivative of F is differentiable on the entire domain. (c) For the...
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...
please show all work, even
trivial steps. Here are definitions if needed. do not write in
script thank you!
4. Letf: R2 → R2, by f(x,y) = (x-ey,xy). a. Find Df (2,0). b. Find DF-1(f (2,0)) Inverse Function Theorem: Suppose that f:R" → R" is continuously differentiable in an open set containing a and det(Df(a)) = 0, then there is an open set, V, containing a and an open set, W, containing f(a) such that f:V W has a continuous...
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