\[\iiint_V f \left(\nabla \cdot \vec A\right) dV = ?\] Integration by parts is derived from the product rule for derivatives: \[\begin{align*} &\frac{d}{dx} (fg) = f \left(\frac{dg}{dx} \ri...

This is the improper integral we want to look at: \[I = \int_a^\infty f(x) dx.\] I will show that for the integral to coverge (exist), \(f(x)\) must decline faster than \(1/x\) for large \(x\). (...

I want to know \[\nabla^2 (\frac{1}{|\vec r - \vec r'|}) = \nabla \cdot \left( \nabla (\frac{1}{|\vec r - \vec r'|}) \right).\] Method 1 — Cartesian Coordinates Let’s first look at \(\nabla (\f...

The Problem Here’s an audio I got. It played normally on my PC, but sounded broken when playing on my phone. Try to play it on your devices and see if you encounter the same issue as mine. Why d...

Transparency A video with an alpha channel is convenient when we’re doing some clip editing. For me, I’d like to use that for karaoke videos, which has a karaoke animation video source on the top l...