Consider a sphere of radius $a$ that is heated uniformly and then immersed in cold water. The temperature in the sphere can be modelled by $\theta(r,t)$ where $\theta(r,t)$ satisfies $$\frac{\partial \theta}{\partial t}= \frac{D}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial \theta}{\partial r} \right)$$ where $0<r<a, t>0, \theta(a,t)=0,\theta(r,0)=1$. I’m given that the solution is $$\theta(r,t)=\frac{1}{r}\sum_{n=1}^{\infty}a_ne^{\left(\frac{\pi n}{a}\right)^2Dt}\sin\left(\frac{\pi n}{a}r\right)$$ where $$a_n=\frac{-2a(-1)^n}{\pi n}$$ I’m […]

I’m studying physics, so I’m sorry if I’ll write some inexact things in this post. I wish you can understand me. If we have 1D wave equation: $$\frac{\partial^2 \psi}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2}$$ we say that it’s always possible (for physics situations…) to decompose the generic solution $f(x+ct)+g(x-ct)$ using Fourier Transforms in $e^{i\omega(x/c\mp t)}.$ But the […]

I’m taking a course in continuum mechanics. Our book is Continuum Mechanics by Reddy, a Cambridge edition. In the second chapter he introduces tensors and defines them to be polyadics. He is specifically concerned with dyadic which are tensors of rank (2,0). In his presentation of dyadic, the notation is introduced, then some properties are […]

A discrete one-dimensional model of optical imaging looks like this: $$I(r) = \sum_i e_i P(r – r_i)$$ Here, the $e_i$ are point light sources at locations $r_i$ in the object and $P$ is a point spread function that blurs each point. We can assume that $P$ is even, non-negative and has a finite extent, ie […]

I’m writing a sort of physical simulator. I have $n$ bodies that move in a two dimensional space under the force of gravity (for instance it could be a simplified version of the solar system). Let’s call $m_1, \dots, m_n$ their masses, $(x_1, y_1), \dots (x_n,y_n)$ their positions, $(vx_1, vy_1), \dots (vx_n,vy_n)$ their velocities and […]

I’ve seen $d(xy) = y(dx) + x(dy)$, but I don’t understand the principle behind it and memorizing it is lame. Can anyone explain what is going on here? For example from physics, $$F = {{dP} \over {dt}}$$ $$F = {{d(mv)} \over {dt}}$$ $$F = {{v(dm) + m(dv)} \over {dt}}$$ Since $$dm = 0$$ $$F = […]

Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here’s what I’ve gotten: $$T_\epsilon\Psi(\mathbf r)= e^{i \mathbf{\epsilon} P/ \hbar}\Psi(\mathbf r)=\sum^\infty_{n=0} \frac{(i\epsilon \cdot (-i\hbar \nabla)/\hbar)^n}{n!} \Psi(\mathbf r)=\sum^\infty_{n=0} \frac{(\mathbf \epsilon \cdot \nabla)^n}{n!}\Psi(\mathbf r)= \Psi(\mathbf r) + (\epsilon \cdot \nabla) \Psi(\mathbf r) + \frac{(\epsilon \cdot \nabla)^2 \Psi(\mathbf r)}{2} + […]

The question that I have to solve is found below. However, I do not know how to start the solution since I am unsure about the defintion of a Gradient/Hamiltonian System. What must I check first to know whether it is a Gradient or Hamiltonian System? Which of the following is gradient/Hamiltonian( Conservative) system or […]

OK, we’ve been asked to derive the equations of motion in spherical coordinates. According to the assignment, we should end up with this: $$ \bf \vec{v} \rm = \frac{d \bf \vec{r} \rm}{dt} = \dot{r} \bf \hat{r} \rm + r \dot{\theta}\hat{\boldsymbol \theta} \rm + r \dot{\phi}\sin \theta \bf \hat{\boldsymbol\phi}\rm $$ In this case θ is the […]

Related to this question. Note that I’m using the geometer definition of an $n$-sphere of radius $r$, i.e.$ \\{ x \in \mathbb{R}^n : \|x\|_2 = r \\} $ Suppose I have an $n$-sphere centered at $\bf 0$ in $\mathbb{R}^n$ with radius $r$ which has been divided into $2^k$ orthants by $k$ axis-aligned hyperplanes (note, $k […]

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