Reference no: EM132252065

Differential Geometry -

Exercise 1 - Due to rotation of the sphere

K = {(0, y, z) ∈ IR³ |(y-2)² + z² = 1} ⊂ R³

and the z axis appears a surface T which is called a focus. Construct a differentiable function f : IR³ → IR with regular value O and T = f^-1(0). Prove that f cannot be chosen without singularities.

Exercise 2 - Show that the equation y(x-1) + zx = 0 describe a regular surface E ⊂ IR³. Construct an imbedding u : IR²→IR³ with u(IR2) = E.

Exercise 3 - Construct a differentiable function g : IR² → R with a finite number of critical points, regular value O and g^-1(0) = {(x, y) ∈ IR²|X (x - 1) = 0}.

Complex Analysis -

Exercise 1 - (a) Prove or refute: Is f : Ω → ⊄ holomorphic, whereas Ω ⊂ ⊄ is an area and |f(2)| = constant in Ω then f is constant.

(b) Observe a holomorphic function f : Ω → ⊄, Ω is open with αRef + βImf + γ = 0 for real constants α, β, γ with α² + β² ≠ 0. Show f is constant.

Exercise 2 - f, g : Ω → ⊄ holomorphic and γ a curve in Ω with initial point A and endpoint B show that

∫_γf'(z)g(z) dz = f(B)g(B) - f(A)g(A) = ∫_γf(z)g'(z) dz

Attachment:- Assignment Files.rar