Recent content by William Crawford

The Laplacian, or vector Laplacian to be precise, is naturally defined for vector functions as well. In the framework of vector calculus this is often done as $$ \Delta\mathbf{v} = \nabla(\nabla\cdot\mathbf{v}) - \nabla\times(\nabla\times\mathbf{v}), $$ which has the added benefit of being valid...

Hi thanks for your respons. Isn't the cross-product of two matrices an ill defined notion? At least, I won't call it a "standard" opperation.

Suppose ##A = A_i\mathbf{\hat{e}}_i## and ##B = B_i\mathbf{\hat{e}}_i## are vectors in ##\mathbb{R}^3##. Then \begin{align} \Delta\left(A\times B\right) &= \epsilon_{ijk}\Delta\left(A_jB_k\right)\mathbf{\hat{e}}_i \\ &= \epsilon_{ijk}\left[A_j\Delta B_k + 2\partial_mA_j\partial_mB_k + B_k\Delta...

Hi Physics Forums, Quick question! Are every automorphism on a vectors space ## V ## over some field ## \mathbb{F} ## nothing but the identity mapping in disguise? The reason for asking is; automorphisms are (from my point of view) basically a change of basis, and vectors are invariant under...

Never mind, I've solved it myself. Simply using that $$ \min_{\lbrace s_n\rbrace}\mathcal{H} = \sum_n\min\left(-Js_ns_{n+1}\right). $$

Hi, I know that the ground state of the spin-1/2 Ising model is the ordered phase (either all spin up or all spin down). But how do I actually go about deriving this from say the one-dimensional spin hamiltonian itself, without having to solve system i.e. finding the partition function? $$...

The action potential satisfy a 2nd order ODE according to the Hodgkin-Huxley model. See equation (30) of their seminal paper [1]. This equation has no closed form solution, but can be solved numerically. [1]. HODGKIN AL, HUXLEY AF. A quantitative description of membrane current and its...

Nonlinear dynamics and chaos by Steven Strogatz is a classic and often the book used in introductory courses on the subject. It's exceptional well written and easily digestible. More advanced treatments of the subject depends on the direction you want to go in. Chaos is a big field with a lot of...

I would expect the equation to be $$ \bigg(\frac{dR}{dt}\bigg)^2 = \frac{GM}{R} $$ if it where to describe the gravitational collapse of a non-relativistic star, as a fluid parcel located at distance ##R## is gravitational bound (the Viral theorem). But I might be wrong, I haven't really studied...

No worries! It was me that read your original post in a hurry. Your differential equation in ##y^\prime## belong to a notorious difficult class of ODE's called Abel's nonlinear ODE's of the fist-kind. I haven't had the change nor time to study this class of ODE's, so I'm afraid that I can't...

Your ODE is a second-order linear equation with constant coefficients. It is rather straightforward to solve, simply observe that you can write it in the following form $$ \big(e^{kt}y^\prime\big)^\prime = - ae^{kt}.$$ Now you simply have to integrate twice.

Let ##c## be a rational number (possibly irrational), then there exist a sequence ##\{q_n\}_{n\in\mathbb{N}}## of purely rational numbers that converges to ##c##. Therefore (assuming ##f## is continuous) $$ \begin{align*} f(cx) &= \lim_{n\rightarrow\infty}f(q_nx) \\ &=...

The function ##f(x) = \mathrm{max}(x,0)## (i.e. the positive part) is also a solution to Cauchy as well and though it is continuous it isn't differentiable at ##x=0## and thus not analytic. Check for yourself if also satisfy the original functional equation.

I believe that ##f(x) = ax## for ##a\in\{0,1\}## are the entire family of continuous solutions (consisting of elementary functions) to the original functional equation. As any continuous solution also has to be a solution to the Cauchy's functional equation. However there exist non-continuous...

True, however that just imply that either ##a=1## or ##a=0##.