Define φ : Z[ √ 2] → Z[ √ 2] defined by φ(a + b √...

Define φ : Z[ √ 2] → Z[ √ 2] defined by φ(a + b √ 2) = a − b √ 2 for all a, b ∈ R is an automorphism of rings.

Expert Solution

Colby Messinger answered 2 years ago

Given a function φ(z) with z = x+iy let U(x, y) = ½ [φ(x+iy) +...

Given a function φ(z) with z = x+iy let U(x, y) = ½ [φ(x+iy) + φ(x-iy)] and V(x, y) = i/2 [φ(x+iy) –φ(x-iy)] A) For φ(z) = z2 find U and V and their induced vector fields E =▼U and F =▼V also show that ▼2U = ▼2V = 0 B) Repeat for f(z) = z3 C) For f(z) = ln z we get U(x, y) = ½ ln (x2+y2) and V(x, y) = arctan (y/x) Find ▼U (electrostatic...

Let Z[ √ 2] = {a + b √ 2 | a, b ∈ Z}. (a)...

Let Z[ √ 2] = {a + b √ 2 | a, b ∈ Z}. (a) Prove that Z[ √ 2] is a subring of R. (b) Find a unit in Z[ √ 2] that is different than 1 or −1.

For x = ρsinφcosθ; y = ρsinφsinθ; z = ρcosφ: a/ Express ∂^2/∂y^2 with ρ, φ,...

For x = ρsinφcosθ; y = ρsinφsinθ; z = ρcosφ: a/ Express ∂^2/∂y^2 with ρ, φ, θ, and their partials. b/ Express ∂^2/∂z^2 with ρ, φ, θ, and their partials. c/ Express the Laplacian operator using spherical coordinates.

Let ∼ be the relation on P(Z) defined by A ∼ B if and only if...

Let ∼ be the relation on P(Z) defined by A ∼ B if and only if there is a bijection f : A → B. (a) Prove or disprove: ∼ is reflexive. (b) Prove or disprove: ∼ is irreflexive. (c) Prove or disprove: ∼ is symmetric. (d) Prove or disprove: ∼ is antisymmetric. (e) Prove or disprove: ∼ is transitive. (f) Is ∼ an equivalence relation? A partial order?

Define the following order on the set Z × Z: (a, b) < (c, d) if...

Define the following order on the set Z × Z: (a, b) < (c, d) if either a < c or a = c and b < d. This is referred to as the dictionary order on Z × Z. (a) Show that there are infinitely many elements (x, y) in Z × Z satisfying the inequalities (0, 0) < (x, y) < (1, 1). (b) Show that Axioms O1–O3 ( Trichotomy, Transitivity, Addition for inequalities) are satisfied for this...

Define a relation ~ on Z x Z such that (a,b) ~ (c,d) precisely when a...

Define a relation ~ on Z x Z such that (a,b) ~ (c,d) precisely when a + b = c + d. Let R = {[(a,b)] : (a,b) in Z x Z} (i.e. R is the set of all equivalence classes of Z x Z under the equivalence relation ~). For each of the following operations, determine whether or not the operation is well defined. Prove your answer. [(x,y)] * [(w, z)] = [(x + w, y + z)] [(x,y)]...

Given a scalar fifield φ(x, y, z)=3x2-yz and a vector field F(x, y, z)=3xyz2+2xy3j-x2yzk. Find: (i)F.∇φ....

Given a scalar fifield φ(x, y, z)=3x2-yz and a vector field F(x, y, z)=3xyz2+2xy3j-x2yzk. Find: (i)F.∇φ. (ii)F×∇φ. (iii)∇(∇.F).

Show that the relation 'a R b if and only if a−b is an even integer defined on the Z of integers is an equivalence relation.

Let f : Z × Z → Z be defined by f(n, m) = n −...

Let f : Z × Z → Z be defined by f(n, m) = n − m a. Is this function one to one? Prove your result. b. Is this function onto Z? Prove your result

Let S = {2 k : k ∈ Z}. Let R be a relation defined on...

Let S = {2 k : k ∈ Z}. Let R be a relation defined on Q− {0} by x R y if x y ∈ S. Prove that R is an equivalence relation. Determine the equivalence class

Question