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The beauty of Laplace Equation

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Md.Shahjalal Talukder:
PHYSICS HAS ITS own Rosetta Stones. They’re ciphers, used to translate seemingly disparate regimes of the universe. They tie pure math to any branch of physics your heart might desire. And this is one of them:

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It’s in electricity. It’s in magnetism. It’s in fluid mechanics. It’s in gravity. It’s in heat. It’s in soap films. It’s called Laplace’s equation. It’s everywhere.

Laplace’s equation is named for Pierre-Simon Laplace, a French mathematician prolific enough to get a Wikipedia page with several eponymous entries. In 1799, he proved that the the solar system was stable over astronomical timescales—contrary to what Newton had thought a century earlier. In the course of proving Newton wrong, Laplace investigated the equation that bears his name.

It has just five symbols. There’s an upside-down triangle called a nabla that’s being squared, the squiggly Greek letter phi (other people use psi or V or even an A with an arrow above it), an equals sign, and a zero. And with just those five symbols, Laplace read the universe.

Phi is the thing you’re interested in. It’s usually a potential (something physics majors confidently pretend to understand), but it can be plenty of other things. For now, though, let’s say that it represents the height above sea level of every point on a landscape. On a hilltop, phi is large. In a valley, it’s low. The nabla-squared is a set of operations collectively called the Laplacian, which measures the balance between increasing and decreasing values of phi (heights) as you move around the landscape.

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