Serie 3
... Serie 3 d-infk Prof. Dr. Emmanuel Kowalski Analysis II Serie 3 ETH Zürich HS 2019 3.1. Which of the ... integers and a2 + b2 + c2 < 2019}; ( c) C = {(x, f(x)) ∈ R2 |x ∈ (0, 1], f(x) = sin( 1x)}; (d) D = {(cos θ ... Serie 3 d-infk Prof. Dr. Emmanuel Kowalski Analysis II Serie 3 ETH Zürich HS 2019 3.1. Which of the ... integers and a2 + b2 + c2 < 2019}; ( c) C = {(x, f(x)) ∈ R2 |x ∈ (0, 1], f(x) = sin( 1x)}; (d) D = {(cos θ ... Serie 3 ...
Serie 3
... . Which of the following sets are compact? (a) A = {(x, y) ∈ R2 |x2 + y2 < 2019}; (b) B = {(a, b, c) ∈ R3 ... | a, b, c are integers and a2 + b2 + c2 < 2019}; ( c) C = {(x, f(x)) ∈ R2 |x ∈ (0, 1], f(x) = sin( 1x ... . Which of the following sets are compact? (a) A = {(x, y) ∈ R2 |x2 + y2 < 2019}; (b) B = {(a, b, c) ∈ R3 ... | a, b, c are integers and a2 + b2 + c2 < 2019}; ( c) C = {(x, f(x)) ∈ R2 |x ∈ (0, 1], f(x) = sin( 1x ... Serie 3 ...
Solution 3
... dx for any ϕ ∈ C∞ c (I). Hence, g ∈ Lp(I) is indeed the weak derivative of u ∈ Lp(I) and u ∈ W 1,p(I ... (†) found in (a). By ( 3), we have in particular, ∀ϕ ∈ C∞ c (I) : − ∫ I gu′ϕ′ dx = ∫ I (hu− f)ϕdx. Hence, the ... dx for any ϕ ∈ C∞ c (I). Hence, g ∈ Lp(I) is indeed the weak derivative of u ∈ Lp(I) and u ∈ W 1,p(I ... (†) found in (a). By ( 3), we have in particular, ∀ϕ ∈ C∞ c (I) : − ∫ I gu′ϕ′ dx = ∫ I (hu− f)ϕdx. Hence, the ... Solution 3 ...
Solution 3
... dx for any ϕ ∈ C∞ c (I). Hence, g ∈ Lp(I) is indeed the weak derivative of u ∈ Lp(I) and u ∈ W 1,p(I ... part (a). By ( 1), we have in particular ∀ϕ ∈ C∞ c (I) : − ∫ I u′ϕ′ dx = ∫ I (u− f)ϕdx. Hence, the ... dx for any ϕ ∈ C∞ c (I). Hence, g ∈ Lp(I) is indeed the weak derivative of u ∈ Lp(I) and u ∈ W 1,p(I ... part (a). By ( 1), we have in particular ∀ϕ ∈ C∞ c (I) : − ∫ I u′ϕ′ dx = ∫ I (u− f)ϕdx. Hence, the ... Solution 3 ...
Serie 3
... transversality condition with the initial value u(s, s) = s. What is occurring in this case? ( c) Define w1 := x ... Serie 3 d-math Prof. M. Iacobelli Analysis 3 Serie 3 ETH Zürich HS 2021 3.1. Characteristic method ... transversality condition with the initial value u(s, s) = s. What is occurring in this case? ( c) Define w1 := x ... Serie 3 ... Serie 3 ...
Hints 3
... 2021 3.1. A closedness property (a) Distinguish the cases 1 < p <∞ and p =∞. In the first case apply ... (I). ( c) Solve for v = u+ v0, where u is the the solution from part (b) with a suitable right and ... 2021 3.1. A closedness property (a) Distinguish the cases 1 < p <∞ and p =∞. In the first case apply ... (I). ( c) Solve for v = u+ v0, where u is the the solution from part (b) with a suitable right and ... Hints 3 ...
Hints 3
... 2018 3.1. A closedness property (a) Distinguish the cases 1 < p < ∞ and p = ∞. In the first case apply ... (I). ( c) Solve for v = u + v0, where u is the the solution from part (b) with a suitable right and ... 2018 3.1. A closedness property (a) Distinguish the cases 1 < p < ∞ and p = ∞. In the first case apply ... (I). ( c) Solve for v = u + v0, where u is the the solution from part (b) with a suitable right and ... Hints 3 ...
Serie 3
... that M = U�0 . c) Use a) and b) to prove ( 1). 3. Consider the vector fields X, Y, Z on S2 given by X(p ... flow of Y for t ∈ R. Prove that ϕ− 1 ◦ ψt ◦ ϕ is the flow of ϕ∗Y for t ∈ R. c) Let ϕ be a diffeomorphism ... = U0 . c) Use a) and b) to prove ( 1). 3. Consider the vector fields X, Y, Z on S2 given by X(p) = ξ × p ... flow of Y for t ∈ R. Prove that ϕ− 1 ◦ ψt ◦ ϕ is the flow of ϕ∗Y for t ∈ R. c) Let ϕ be a diffeomorphism ... Serie 3 ...
Serie 3
... (Simply rotate the sphere by an element of SO( 3) sending ξ to e3 and use Exercise 4 c) ). Thus from ... on M and let ψt be the flow of Y for t ∈ R. Prove that ϕ− 1 ◦ ψt ◦ ϕ is the flow of ϕ∗Y for t ∈ R. c ... by an element of SO( 3) sending ξ to e3 and use Exercise 4 c) ). Thus from exercise 1, we already have ... for t ∈ R. Prove that ϕ− 1 ◦ ψt ◦ ϕ is the flow of ϕ∗Y for t ∈ R. c) Let ϕ be a diffeomorphism on M ... Serie 3 ...
Serie 3
... f2 + 3 f3 +4 f4 x+ 1 +− 1 +0 +32 h1 h2 Figure 1: The counterexamples for 3.1 (b) and (e). 3.2. An ... contradicts A ∩ A{ = ∅. ( c) Let d be the Euclidean distance. Since [0, 1] \ {q} is open in ([0, 1], d) for any ... f2 + 3 f3 +4 f4 x+ 1 +− 1 +0 +32 h1 h2 Figure 1: The counterexamples for 3.1 (b) and (e). 3.2. An ... contradicts A ∩ A{ = ∅. ( c) Let d be the Euclidean distance. Since [0, 1] \ {q} is open in ([0, 1], d) for any ... Serie 3 ...
