Real Analysis : Finding a Maximum using Lagrange Multipliers

Please see the attached file for the fully formatted problem. What is the maximum of F = x1 +x2 +x3 +x4 on the intersection of x21 +x22 +x23 + x24 = 1 and x31+ x32+ x33+ x34= 0? As this is an analysis question, please be sure to be rigorous and as detailed as possible.

Poincare's Lemma and its Converse

Please see the attached file for the fully formatted problem. For  phi E C2[R3 ! R3], curl grad phi = 0. Prove this. The converse is ”Poincare’s Lemma”: if f E C1[R3 --> R3] and if curl f = 0, then f is a gradient, i.e., f = grad  for some  2 C2. Try it this way: if f = grad phi, then phi (x1, x2, x3) = phi(0)+ .... See why? Be that as it may, this function phi is perfectly well - defined. So start from scratch with this phi taken out of the air and see what it’s gradient is, assuming curl f = 0.

Laplacian : Grad and Curl Proof

Please see the attached file for the fully formatted problems. Show that for f E C2(R3 --> R3), grad x curl =grad(div f) - DELTA f

Critical Point : Non-Degenerate

Please see the attached file for full problem description. Show that f(x) = x1x2 + x2x3 + x3x1 has a non - degenerate critical point at x = 0 and describe the shape of f as concretely as possible.

Jacobian Matrix of a function and it's Inverse

Please see the attached PDF file. Thanks!