Have you ever wondered why the Laplacian is so ubiquitous in physics and engineering? I’ve heard many explanations, many of which refer to symmetry considerations, but there is one argument that has to do with the intuitive meaning of the Laplacian itself.

Consider the heat equation:

The left-hand side tells us how quickly the temperature is changing at a given point 𝑥⃗. What would we expect this to depend on? Well, it seems reasonable to think that it would depend on the temperature of the surrounding area. If the neighborhood around 𝑥⃗ is, on average, hotter than at 𝑥⃗, then we would expect the temperature at 𝑥⃗ to rise. Conversely, if the neighborhood is colder than at 𝑥⃗, we would expect the temperature to fall. And that’s exactly what the Laplacian is telling us!

In the one-dimensional case, the Laplacian is simply:

As you may know from calculus or from this article , the second derivative can be expressed as a limit:

If we rewrite the term under the limit, we see that this is proportional to:

This tells us that the second derivative is (proportional to) the difference between the value of the function at the point of interest, 𝑇(𝑥), and the average of the values in the immediate neighborhood. The same argument applies to the multidimensional case. So now we have an intuitive understanding of what the Laplacian means and why it appears in so many physical equations, such as the heat equation.

Pythonic Scientist

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It’s fascinating how a concept that arises from simple calculus operations can have such wide-ranging applications in physics and engineering. One thing to note is that the Laplacian also arises in other physical equations, such as the wave equation and the Schrödinger equation in quantum mechanics. In each case, the Laplacian represents a measure of how the system is changing or evolving over time with respect to its surroundings.