
Question 1 (15 marks) Let f be a holomorphic, non-constant function on a domain 2 c...
4. Let p(u, v) be a non-zero Cl function of two real variables whose gradient is non-zero on the set fp 0, and let f u+ iv be holomorphic on region 2 C C and satisfy p(Re (f), Im (f))-0. Prove that f is constant on Ω. Conclude as special cases that if f is holomorphic on a connected open set and f is real valued, then f is constant, or if the modulus off is constant on Ω, then...
Prove the following Theorem:
Theorem. f : X → R is continuous + for any open set U C R, the pre-image f(U) is open in the domain of X (i.e., f(U) = XnV for some open set V C R).
Please answer this question
Implicit Function Theorem in Two Variables: Let g: R2 - R be a smooth function. Set Suppose g(a, b)-0 so that (a, b) є S and dg(a, b) 0. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above (2) Since dg(a, b)メ0, argue that it suffices to assume a,b)メ0. (3) Prove the...
1. Let k E C and define f:C+C to be the constant function f(x) = k. Use part (1) or (2) of to prove that f is continuous. (1) For every closed set FCC, the preimage f-1(F) is F' n D for some closed set F' (2) For every open set G C C, the preimage f-1(G) is G'' n D for some open set G'.
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
2. Let V and W be vector spaces over F. Define the set v, w) |v V andwEW This is called the product of V and W (a) Show that V x W is a vector space. (b) Define a map w : V → V × W by w (z) = (z,0). Show that w is an injective linear map. Note that we can define a similar map lw (c) If (d) Show that V x W. (V W...
surface patch for S. regular surface and f: S Ra smooth EXERCISE 3.44. Let S be a function. Assume that the point p e S is a critical point of f, which means that dfp(v) 0 for all v e TpS. Define the Hessian of f atp in the direction v as Hess(f)p(v) (foy)"(0), where y is a regular curve in S with y(0) = p and y'(0) = v. Prove that the Hessian is well defined in the sense...
08. (3+2+1+1=7 marks) Let (E, d) be a metric space and let A be a non-empty subset of E. Recall the distance from a point x e E to A is defined by dx, A) = inf da, a).. a. Show that da, A) - dy, A) <d(x,y)Vxy e E. Let U and V be two disjoint and closed subsets of E, and define f: E- dz,U) R by f(x) = 0(2,U) + d(«,V) b. Show that f is continuous...
this is an optimization subject.
that is example 2.33
Question 2 (6 Marks) (Chapter 2) Consider the function f : R3 -R defined as f(x1,2,3 +4eli++21), (G) Explain why f has a global minimum over the set Hint: Read Example 2.33 (i) Find the global minimum point and global minimum value of f over the set C. Example 2.33. Consider the function/(x1,x2)=xf+xỈ over the set The set C is not bounded, and thus the Weierstrass theorem does not guarantee the...