Please tell me if the following (see attached) series converge. This is a problem from my text book and unfortunately there are no solutions for this problem. I need justification, not just a yes it converges or no it doesn't answer.
Please see the attached file for the fully formatted problems. The sphere x20 + x21 +· · ·+x2n of radius sqrt(n) is eqiupped with its natural area, normalized so that the total area is unity. Show that the normalized volume of the spherical band where a a<=x0 <= b is ..... and prove that lim... Hint: 1 − x <= e^(& ...continues
Matrix Series : Trace Log and Log Det
Please see the attached file for the fully formatted problems. Let K be an nxn matrix and a small number. Imitating ..... valid for small x, it is natural to define .... Explain why this makes sense. Prove trace log = log det(I + K) Still with small so that everything makes sense. Hint: What is I − K +2K2 ...continues
Please see the attached file for the fully formatted problems. Prove that if D is the closed disc |x| =< 1 in R2, then any map f E C2[D --> D] has a fixed point: f(x) = x. The proof is by contradiction, and uses Stokes theorem. Follow the steps outlined below. (1) Define a new map F(x) = ... ..... Show that F has no fixe ...continues
Mean Value Theorem for Harmonic Functions : Green's Identity
Please see the attached file for the fully formatted problems. Let h 2 C2(R3) be harmonic (h = 0). Use Green’s identity for .... to show that ...is independent of the value of R. Then deduce the mean value theorem .... Now what can you say if limx!1 h(x) = 0?
Euler Equation of Hydrodynamics
Start from Euler’s equation ... Take the divergence to obtain delta-p as a quadratic expression in ... Use divv = 0 to make this as simple as you can. Assuming v and p vanish sufficiently rapidly at 1, express p itself in terms of v alone.
Harmonic Functions : Green's Identity and Mean Value Theorem for Harmonic Functions
Please see the attached file for the fully formatted problems. Let h 2 C2(R3) be harmonic (h = 0). Using Green’s identity for .... is independent of the value of R. Then one can deduce the mean value theorem .... Now what can you say if limx!1 h(x) = 0?
Volume of a Sphere : Normalized Volume of Spherical Band
Please see the attached file for the fully formatted problems. Consider the sphere x20 + x21 + · · · + x2n = n of radius sqrt(n). Show that the normalized volume of the spherical band where a <= x0 <= b is .... Hint: 1 − x =< e^−x will be helpful at one point.
Please see the attached file for the fully formatted problems. Let K be an nxn matrix and .. a small number. Imitating .... valid for small x, it is natural to define .... Explain why this makes sense. Prove trace log... = log det.... Still with ... small so that everything makes sense.
Show that the 2-cycle of the quadratic map is stable if 3/4<5/4. The quadratic map is x(n+1)=x(n)^2-c.
