What Is the Laplacian?

[harleyspeedthrust]

Love the tensor calculus teaser! It's professors like you who get students curious and interested in math by asking meaningful questions. It certainly took me a lot longer than it should have to wonder why I needed a background Cartesian coordinate system for everything and whether there was a way to work with a coordinate system on its own terms. Your tensor calculus playlist (and book) has answered this question for me and I cannot thank you enough.

[Nickle]

I really do like the emphasis on meaning. Combine that with the drive to more abstraction and its really refreshing.

[SomeMrMindism]

For those who are wondering: take a scalar function (like u()), use the exterior derivative to transform it to a 1-form, use the metric tensor to transform it to a vector. This is the gradient of the function. Then use the metric tensor to get a volume form, use the gradient to contract the volume form to a 2-form, then use the exterior derivative again to get a new 3-form. The coefficient of this 3 form is the Laplacian (in R^3). This object is indifferent to change of coordinates. IF the metric is cartesian, then all this procedure reduces to the divergence of the gradient

[georgesadler]

Professor MathTheBeautiful, thank you for a deep explanation and introduction to the Laplacian in Partial Differential Equations. This is an error free video/lecture on YouTube.

[anonymous_]

Cannot express how thankful I am for this. Was wondering about the geometrical interpretation of the laplacian for days. Finally I have the answer.

[suboptimallife]

Did the drum meet Descartes? Hilarious.

[AbhishekSachans]

Your explanations are so insightfully excellent. Your teaching style is one of its kind. I'm fortunate to have found you before I begin my college journey. Excited there is so much to learn from you. Thanks for a good impact.
Greetings from India!

[Malletman]

The calculation in the latter half is a nice one! It's exactly the calculation that gives you the ability to determine the value of a point under a harmonic function by taking a contour around it, since a function is harmonic iff the Laplacian is zero.

[khashtidattpandey]

You are really great sir, nowhere I could find such a nice explanation..., thanks a lot

[scitwi]

17:06 Actually, a surface can have a "shape" as you called it (be curved) and still have a zero Laplacian ;> An example of that is a saddle point on a hyperbolic paraboloid (the shape of a saddle). At that point, the surface bends up in one direction, but it also bends down in the other (perpendicular) direction, so on average the curvatures cancel.
17:54 To confirm that you are an Illuminatus? :>
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19:51 What if my drum has a shape of a violin or guitar box? :q Is there a way to come up with a fancy coordinate system for that?

[scitwi]

05:00 So why is it that when we change the coordinate system, the set of solutions that makes the basis for other solutions (e.g. the "normal modes" for the membrane) also changes?

[JJC]

I love the way you move when you move really fast.

[sergejbojanovic]

Thank you for the amazing effort you're putting into making such a complex subject as PDEs really understandable, Dr Grinfeld!
This video was when the basic equations (Laplace's, heat, wave) started clicking for me.

I am, however, curious what would we get if, instead of computing average value of the function on the boundary of the point neighborhood we did it over the entire neighborhood.
E.g. for
1) 1D, average on the line segment instead of average of its endpoints
2) 2D, average on the entire circle instead of only its boundary
3) 3D, average on the ball, instead of only the sphere

Is this value somehow related to the Laplacian?
What would be its physics interpretation?

[Hythloday]

Good little sales pitch in there for Tensors ... lol. Is it sign up time for next years courses already ;o)

[hogun]

This is exactly what I've been looking for thx.

[davidkwon]

Your lecture is really beautiful. Thank you!!!

[wowlikefun]

Thanks! The lecture is really well prepared, bravo!

[klammer]

The descent into madness is almost complete ;-)

[duckymomo]

I always mix up the symbols laplacian and gradient

What if the triangle is gradient squared (upside down triangle squared)?

[structuralanalysis]

Sometimes, you sound like our mentor: Richard Feynman. Love your work.