Recent content by Luke Tan

In section 1.12 Variational Approach to the Solution of the Laplace and Poisson Equations, Jackson mentions that in electrostatics, we can consider "energy type functionals". He gives, for Dirichlet Boundary Conditions, $$I[\psi]=\frac{1}{2}\int_{V}\nabla\psi\cdot\nabla\psi d^3x-\int_{V}g\psi...

yea I know that it's really hard, but now I feel like I'm lacking equations (the solution to the poisson equation isn't unique, unless boundary conditions are specified, and in this case they are not - I am trying to find the boundary conditions from the solution), and so I can't even solve it...

By "solving the full BVP of the poisson equation", do you mean inside the cavity only? I'm not sure how we can do that, since we need the boundary conditions to get a unique solution to the Poisson equation, but I'm trying to do it in reverse here.

Let us say we have a cavity inside a conductor. We then sprinkle some charge with density ##\rho(x,y,z)## inside this surface. We have two equations for the electric field $$\nabla\times\mathbf{E}=0$$ $$\nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_0}$$ We also have the boundary conditions...

I forgot to attach the given diagram, here it is (the axle is vertical, thus the plate is vertical)

My struggle here comes from finding the bending moment ##N(x)##. My working is as follows. We want to find the bending moment on an element a distance ##x## away from the axis of rotation. To do so, let us consider the bending moment due to the force on an element ##\xi>x## away from the axis...

What options do you use to make a simple line plot look good?

So would it be correct to say that it is the numerical value that is invariant?

Hey guys, so recently I've been trying to use mathematica to plot graphs for my scientific papers, and I've been starting to wonder - what plot options do you guys use? How do you plot your graphs in mathematica such that they look presentable in a scientific paper?

Um sorry I'm not familiar with differential geometry, is there any other way I can understand this?

u and v arent really any definite functions, I just want to get an idea of how the poisson bracket transforms under a canonical transformation and what exactly is invariant. Classical poisson bracket

I've recently been starting to get really confused with the meaning of equality in multivariable calculus in general. When we say that the poisson bracket is invariant under a canonical transformation ##q, p \rightarrow Q,P##, what does it actually mean? If the poisson bracket ##[u,v]_{q,p}##...

oh so this is a special case, and does not disqualify differentiating and setting to 0 as a general method of finding the peak frequency?

In a numerical Fourier transform, we find the frequency that maximizes the value of the Fourier transform. However, let us consider an analytical Fourier transform, of ##\sin\Omega t##. It's Fourier transform is given by $$-i\pi\delta(\Omega-\omega)+i\pi\delta(\omega+\Omega)$$ Normally, to find...

I've started reading Goldstein Classical Mechanics recently and I've found the problems inside to be much more difficult than what I'm used to. Before this, I used to read books like David Morin's Introduction to Classical Mechanics, with problems that had extremely detailed solutions and where...