Octupal Integrals: A New Level of Insanity in Mathematics

[Invictious]

$$\int\int\int\int \int\int\int\int f\left(x_{1},x_{2},x_{3},x_{4},x_(5),x_(6),x_(7),x_(8)\right) \ dx_{1} \ dx_{2} \ dx_{3} \ dx_{4} \ dx_{5} \ dx_{6} \ dx_{7} \ dx_{8}$$

Anyone fancy octupal integrals?
Just out of curiosity, when will we ever even NEED to do this?
I was presented a problem in this form today, and it's rather disturbing. Is it even applicable to applied math, or will it fall into the realm of pure mathematics?

 

[Gib Z]

Just fixing up the latex:

$$\int\int\int\int \int\int\int\int f\left(x_1,x_2,x_{3},x_{4},x_5,x_6,x_7,x _8\right) \ dx_{1} \ dx_{2} \ dx_{3} \ dx_{4} \ dx_{5} \ dx_{6} \ dx_{7} \ dx_{8}$$

I would say you would need Cauchy's repeated integral formula, and there would be no point.

 

[jambaugh]

Usually beyond two or three variables one uses vector notation:
$$\int_{\Omega} f(\vec{x})d^n x$$
And geometric properties of the problem to solve them.

These can arise in e.g. statistical mechanics where you have 4 configuration variables and 4 canonical momenta. You integrate a distribution over a region of phase-space to find a probability.

More generally in the functional integral approach to quantum field theory one takes the limit of integrals of an arbitrary number of variables as that number goes to infinity. Each variable (or set of three variables) represents one point on the path of a particle. In the limit one is integrating over all paths and the formal integral is written:

$$\int \mathcal{D}\phi F[\phi]$$
where phi is the variable function expressing the path. I.e. its value for each parameter phi(t) is considered an "independent" variable.

These are difficult to even define as generally meaningful and only used for certain special forms which arise in physics. See:
http://en.wikipedia.org/wiki/Functional_integration"

Regards,
James Baugh

 

[Pseudo Statistic]

Just fixing up the latex:

$$\int\int\int\int \int\int\int\int f\left(x_1,x_2,x_{3},x_{4},x_5,x_6,x_7,x _8\right) \ dx_{1} \ dx_{2} \ dx_{3} \ dx_{4} \ dx_{5} \ dx_{6} \ dx_{7} \ dx_{8}$$

I would say you would need Cauchy's repeated integral formula, and there would be no point.

Ofcourse there would be a point!
How do you think those living in 9-D would feel if you told them they couldn't calculate volumes of arbitrary objects?!

 

[Feldoh]

Ofcourse there would be a point!
How do you think those living in 9-D would feel if you told them they couldn't calculate volumes of arbitrary objects?!

Probably a bit angry.