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Course: Multivariable calculus>Unit 2

Lesson 12: Laplacian

Explicit Laplacian formula

This is another way you might see the Laplace operator written. Created by Grant Sanderson.

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• At , should the n term have a squared 'denominator' and also in the sum notation, is the squared missing?
• Yes. You can infer this from the fact that $\frac{\partial^2 f}{\partial x_N}$ for N = 1 is an element of the series, explicitly being $\frac{\partial^2 f}{\partial x_1}$.
• what is the use of Laplacian formula?
• The Laplacian is used to find out if a point where the partial derivatives are zero is a maximum or a minimum. If the Laplacian is negative at that point, it's a maximum. If it is positive, the point is a minimum.
• why isn't the laplacian used as a second derivative test ?
• Multi-variable calculus is more complicated than the single-variable stuff (although there are parallels and understanding the single-variable stuff gives useful insight)...
Basically, the Laplacian doesn't provide enough information. If you want to know more, keep going on the multivariable course and you'll come to the Hessian matrix... go a bit further and it explains it all.
I know...: i've done it and come back to amend this reply as a result.
• Does the Laplacian operator apply only to scalar-valued functions?
• Yes, as you can only take the gradient of a scalar-valued function, and that is the first step of the Laplacian.
• Isn't the Laplacian shown by ∇²f instead of ∆f ?
• Both are acceptable. See Wikipedia article: "[The Laplace operator] is usually denoted by the symbols ∇ ⋅ ∇, ∇^2 (where ∇ is the nabla operator), or Δ." (https://en.wikipedia.org/wiki/Laplace_operator)
• I think one way to relate the Laplacian formula in 3D and in 2D is to think of the curve of the function, in 2D it's obvious to see that the second derivative at the local highest point is negative, if we cut the 3D graph with a slice(which is parallel to the z-y plane or z-x plane)If we draw the curve of 3D function on the 2D slice(if at the highest point we cut it) the 2D graph will appear to have negative second derivative.
• It will be easier to write
△f=∇.∇f as
△f=∇²f
Its sounds better than the summation! :)