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### Course: Multivariable calculus>Unit 2

Lesson 12: Laplacian

# Laplacian computation example

A worked example of computing the laplacian of a two-variable function. Created by Grant Sanderson.

## Want to join the conversation?

• Does the laplacian have anything to do with the laplace transform other than being made convention by the same mathematician?
• They are unrelated. The Laplace transformation involves integration, complex numbers, and exponential functions. It is used widely in electrical engineering.
The Laplacian, on the other hand, is related to multi-variable derivatives and was first used by dear Mr. Laplace in his studies of celestial mechanics. Notice how the Laplacian relates to peaks and valleys? When I think of peaks and valleys on a surface and celestial stuff, I immediately think of the curvature of space-time, theorized by Einstein (relativity). A black hole, in theory, is basically an infinitely deep valley in the fabric of space-time. Interestingly enough, Mr. Laplace was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse! Just a little trivia that I thought you might find interesting.

In addition, the Laplace equation is directly related to the Laplacian--it's the equation where
∇·∇ F = 0 (where F is a function).
• The further we go, the fewer questions are posted here. Only the strong (or interessted) survive.
• This is not a question, but I wanted everyone to see it: MIT OCW Lecture Notes on Differential Equations are wonderful companions to this playlist/course. Don't use Khan Academy's Differential Equations course because it is not comprehensive, and also the handwriting is terrible. I don't know about anyone else, but I can't take the handwriting. Also, check out 3blue1brown (this guy)'s course on Linear Algebra on youtube. Awesome stuff, that.