mean curvature and 2nd variation

[AntonioLao]

In order to satisfy the minimal and maximal surface conditions, the mean curvature must be zero. Likewise, in order to determine the position of any extremum point, the 2nd variation ([math]\delta^2 F[/math]) must also be zero.

[math]\delta^2 F = \epsilon^2 \sum \frac{\partial^2 F}{\partial u_i \partial u_k}a_i a_k [/math].

By the axiom of choice, the parameter [math]\epsilon[/math] is chosen sufficiently small in order to neglect higher terms of its powers. However, if [math]\delta^2 F = 0[/math] then the parameter disappears anyway (so why need it in the 1st place?) it might just be as well to state directly at the outset that

[math]\sum \frac{\partial^2 F}{\partial u_I \partial u_k}a_i a_k =0 [/math]

Where [math] I,k=1,2,3, \ldots[/math] and [math]a_i , a_k[/math] are the direction cosines. A personal comment follows; this is where the flow of mathematical logic had gone around a useless circuit, which means the same in the end. So, why bother?

[Guille]

Do you mean useless for explaining nature or useless EVEN for math? If a math theorem is useful for math, in that it is a brick of the pyramid, then it is enough, at least for math itself, isn't it?

[AntonioLao]

Do you mean useless for explaining nature

I mean redundancy, superfluous, useless as in needlessness. Disguising the axiom of choice as logical statement and implication. If mathematicians wanted something to be zero, why do they have to go about in a long winded mathematical symbolism to say it?

[Guille]

I mean redundancy, superfluous, useless as in needlessness. Disguising the axiom of choice as logical statement and implication. If mathematicians wanted something to be zero, why do they have to go about in a long winded mathematical symbolism to say it?

Maybe because then it's easier to do algebraic works: if the remove things of the equations from one side to an other, etz, it's alway easier to start with all in one side and the other side a zero, because the zero has no value.

[AntonioLao]

Maybe because then it's easier to do algebraic works:

Heisenberg almost died of high fever trying to work out his development of quantum mechanics by the matrix method, knowing the Poisson's brackets of position and momentum operators do not commute [x,p]-[p,x]>0 classically, we know [x,p]-[p,x]=0 is always true.

[Guille]

Heisenberg almost died of high fever trying to work out his development of quantum mechanics by the matrix method, knowing the Poisson's brackets of position and momentum operators do not commute [x,p]-[p,x]>0 classically, we know [x,p]-[p,x]=0 is always true.

His high fever was probably caused by his mind: he had difficulty, so he wanted to get the sensation of difficulty, so his mind made his body have fever, to feel difficulty. Many of this kind of things are mental.

[AntonioLao]

Many of this kind of things are mental.

My mental fever is to solve the thermonuclear controlled fusion engine before I raise the temperature to the point of no return of my time axis. The angle of the time axis is directly proportional to the temperature, the higher the temperature the longer we live. However, this is a disaster in disguise, since our mental time axis still must be harmonized or synchronized with the other time axes of our entire body. The time axis of the liver, of the heart, of the kidney, of the lung, of the stomach, of the hands, of the feet, of the nose, of the eyes, of the ears, of the mouth, of the bones, etc.

[Guille]

My mental fever is to solve the thermonuclear controlled fusion engine before I raise the temperature to the point of no return of my time axis. The angle of the time axis is directly proportional to the temperature, the higher the temperature the longer we live. However, this is a disaster in disguise, since our mental time axis still must be harmonized or synchronized with the other time axes of our entire body. The time axis of the liver, of the heart, of the kidney, of the lung, of the stomach, of the hands, of the feet, of the nose, of the eyes, of the ears, of the mouth, of the bones, etc.

Basically, I think that this time and space axis thing is very complicated. It has lots of things of which we have spoken about, and still what I know looks terribly empty, as if there were 1,000 more posts that we must do about it.

[AntonioLao]

as if there were 1,000 more posts that we must do about it.

This is where I need to start using tensor notations but as I said before, I hate tensors. I just like working with vectors in contraint one dimensional space-time (one of space and one of time) or two dimensional spacetime without the hyphen.

[Guille]

This is where I need to start using tensor notations but as I said before, I hate tensors. I just like working with vectors in contraint one dimensional space-time (one of space and one of time) or two dimensional spacetime without the hyphen.

I also don't like tensors, they are too complicated as to be used in normal-fluent thinking. I also like vectors, they are way more direct as concepts than tensors, and thus, they are easy to think about.