Recent content by jspectral

Homework Statement Prove that C is a field. Homework Equations Field Axioms. The Attempt at a Solution I know there are 10 field axioms, but is it enough to just say something like: Let a,b,c be in C (a+b) + c = a + (b + c) So satisfies addition associativity, and then go...

Homework Statement Use the mean value theorem to show that if x ∈ ℝ>0 then 0 < ( x + 1)^1/5 − x^1/5 < (5x^4/5)^-1 Homework Equations MVT: f(b) = f(a) + f ' (c)*(b-a) The Attempt at a Solution I can see that (5x^(4/5))^-1 is the differential of x^1/5, but I'm not sure what to let...

Homework Statement Let \mathbf{G}(x,y,z) be an irrotational vector field and g(x,y,z) a C^1 function. Use vector identities to simplify: \nabla\cdot(g\nabla \times (g\mathbf{G})) Homework Equations The '14 basic vector identities' The Attempt at a Solution I tried using the...

Okay so the Eigenvectors for mine are: v_1 = \left(\frac{1 + \sqrt{5}}{2}\ , 1 \right) and v_2 = \left(\frac{1 - \sqrt{5}}{2}\ , 1 \right) So A^k(v_1) = \left(\frac{1 + \sqrt{5}}{2}\ , 1 \right) \frac{1 + \sqrt{5}}{2}\ to the power of k for the non-vector and A^k(v_2) =...

Okay, so I got the eigenvalues of the matrix A as \left(\frac{1 \pm \sqrt{5}}{2}\ \right) Now how do I use those eigenvalues with the power of k in order to obtain an expression for F_(k)?

We haven't been taught eigenvalues/vectors yet. Is there any other method you can think of to do this problem?

Homework Statement Using u_k = \[ \left( \begin{array}{ccc} F_{k+1} \\ F_k \end{array} \right)\] u_0 = \[ \left( \begin{array}{ccc} 1 \\ 0 \end{array} \right)\] A = \[ \left( \begin{array}{ccc} 1 & 1 \\ 1 & 0 \end{array} \right)\] Solve for u_k in terms of u_0 to show that: F_k =...

You mean int(1 + x + x^2 + x^3 + ...) = x + x^2/2 + x^3/3 + ... ?

A few, it's for a 2nd year university subject. We've been using proof by induction a fair bit, but other than that I don't know the actual names for them.

Homework Statement \sum\frac{1}{n2^(n+1)} from 1 to infinity. By the way, that 2 is to the power of (n+1), doesn't show clearly. Homework Equations The Attempt at a Solution I have worked out the first few individual calculations, up to n=6, and i know it approaches ln(2)/2, however I...

I tried this method: V(r) - V(R) = \int_{R}^r E.dr where E = \frac{1}{4\pi\epsilon_{0}}\frac{Q}{r^2} But I only end up with factors of \frac{1}{R} and \frac{1}{r} Am I approaching the problem wrong? Should I be integrating over a different domain? Considering the answer: V(r)...

1. Given a sphere of uniform charge Q, radius R: Find an expression for the electrostatic potential V as a function of r for r ≤ R. Prior proof: E= Q(r)/(4πϵ0)r^2 ,for r≤R where Q(r) is the excess charge in the spherical volume of radius r 2. Relevant equation: V = Q/(4πϵ0)r 3. I tried...