What does the second derivative actually do in math and physics? 

It's a great video, but perhaps the visual and conceptual leap from 1D, a line plotted on a 2D graph, to a 3D scalar field was slightly glossed over? You covered it with the leap from charge density to scalar field potential but maybe just one more slide and line would have smoothed it over :-)

Very nice. I want to say that intuition is one facet one can apply to physics but very tough to apply to mathematics. But your explanation is fantastic where I also thought of many things concerning math intuitively. So, I want to say something about your clip with a point inside of the sphere where you call that point as something tangible. I can hold a sphere in my hands such as a basketball but I can never hold a point because a point has no dimension. So, when I see a point in your video I see a small sphere inside a big sphere which may be very misleading for viewers. A point having no dimension quantitatively is appropriately called x naught because is has no value. So, when you take a limit as dx goes to 0 and once the limit is reached we can only imagine that the limit has been exhausted at point zero qualitatively because at that point there is no dimension. And I have always thought that such points should have a separate notation for something imaginable and not real such as the wave function psi which is not a real wave. So, that's what my intuition tells me about points. Also, in your video you state at 8:58 minute that you showed the 3D case about the second derivative where the first case was in 1D. No, the first case is in 2D because you operate as x and f(x) which means you show a function in 2D displayed on x and y axes.

​@@user-ky5dy5hl4dAgree with you. If I may suggest: Intuition is a guide to imagination of how the reality exists. Imagination is each person's view, and when we all concur using the precision of mathematics, then we are realigning our imagination to reality with precision. And when we accept internally this as TRUE, it becomes our intuitive perception, and an almost perfected view of reality. Then we take another step forward. It is why mathematics is precise, but Intuition is still learning based on existing knowledge.

@@raajnivas2550 Intuition + logic. Agree?

@@user-ky5dy5hl4d Intuition is what idiots use. Look up that word!

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You did an entire physics degree without being shown? Not even in QM? Huh

@@jaw0449 I'm in the same boat actually

@@NormanWasHere452 you should go to your profs and and ask for derivations, then. That, or they’re expecting you to do the derivations on your own. No physics program should ever just give formulas (unless freshman courses)

how it's only been an hour since the vid's upload

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I think that's true if all second derivatives. After all, that's all a laplacian is. If I remember correctly, with scalars there is only one meaningful second derivative, but for vectors, 3 can be formed by permitting curl, div, and grad.

You were forced?

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Most physicists admire Feynman second to only Newton himself. He represents the joyful genius and the spirit of scientific curiosity

As I said in another comment, I saw the same concept in "a treatise on electricity and magnetism" by Maxwell, vol I, pag 29 .... so, I don't think was a feynman's idea.

Look at "a treatise on electricity and magnetism" by Maxwell, vol I, pag 29 .... not Feynman's approach. It was well known before Feynman.

The next step is second quantization - redefining the non-relativistic fixed particle mode to a framework capable of analyzing relativistic many body systems in which the number of particles in a system are no longer fixed. There are quite a few approaches to this, the most common and most utilized framework being quantum field theories appropriate for the different types of fundamental interactions and particle properties. Extending to the Fock space - the Hilbert space completion of the symmetric and antisymmetric tensors in the tensor powers of a single particle Hilbert space is standard to incorporate creation and annihilation operators of quantum states that change the eigenvalues of the number operator by one, analogous to the quantum harmonic oscillator. Something that becomes more important in QFTs. You may have already been introduced to some of the fundamental aspects of this approach, as the natural extension beyond a Junior/Senior undergraduate QM course is the introduction of different QFTs, with particular emphasis on QED.

if the distance an object has travelled in the past dt is less than the distance it will travel in the next dt, it means the object is acccelerating