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## Multivariable calculus

### Course: Multivariable calculus>Unit 2

Lesson 12: Laplacian

# Laplacian intuition

A visual understanding for how the Laplace operator is an extension of the second derivative to multivariable functions. Created by Grant Sanderson.

## Want to join the conversation?

• If gradient means what direction you should walk to get higher, shouldn't it therefore mean the "slope of steepest ascent", not descent?
• Yes I think Grant got it wrong there. Because what he said would contradict the behaviour of vectors near those hills and valleys. (vectors pointing towards hills and outward away from valleys)
• Why do they use Nabla so much in so many different equations in multi-variable calc? It's confusing and hard to remember which variation of Nabla means what. :/
• the Nabla means ==> [∂/∂b1, ∂/∂b2, ∂/∂b3, ..., ∂/∂bn] where bi is the ith vector of the basis making up the vector space F is defined in

normally you just see: [∂/∂x, ∂/∂y, ∂/∂z] because the basis is i, j, k, which are just unit vectors that are all orthogonal to one another. This notation is not to be taken to seriously however, because multiplication is not the same as applying an operator to a function
• I searched countless hours into my mind and into the interent, to define divergence of a Scalar Function. Does it even exist? Finally, found it and it is not customary to call it a divergence of Scalar Function, and rather a Laplacian, which is rather a divergence of its gradient Vector field. Thank you.
• the divergence of a scalar function does not make sense from what I have gathered about what divergence represents. Divergence is an operator applied to a vector field that produces a scalar field. When evaluated at a point P= (P0, P1, P2, ...,Pn), the divergence tells you the density of outward flux from an infinitesimal volume centered at P.
• What is a Laplacian output of a saddle point ?
• I'm posting this as "food for thought" rather than a hard & fast answer, but I think that (for a scalar function f of two scalar variables x,y) it's arbitrary unless the up direction has exactly the same curvature as the down direction, in which case it would be zero.
(1 vote)
• So the Laplacian indicates how much of a local minimum or máximum a point is and the video says that it is the analogous to the second derivative test in single variable calculus. But.. we don't use the Laplacian to find local min/max, we use the Hessian determinant in the second partial derivative test to find local Min, Max.. So.. What is it good for??
• There are many applications, such as in physics (physics is all about differential equations). There are also a bunch of stuff it can be used for in math, chemistry, and computer science.
• At the arrow above being red suggests that it's bigger than the yellow under it so the divergence at this point isn't 0 right?