Partial differntials with hypercomplex numbers

[mrcuteblackie]

Thank you for your knowledge of maths and science, and for the good cooperation.
First of all, may I asked a question on solving partial differential equations using hyper complex variables?

Please can you briefly explain why the constants of integration in solving partial differential equations (x,t) within the 4D hyper-complex domain are not arbitrary functions of x,t, but are actual constants which should not be differentiated when finding derivatives.
If we have three constants for 1 initial + 2 boundary conditions, and we want to set t=0, then we must leave x as it is. Therefore the arbitrary constants are instead arbitrary functions which depend on either x or t. Therefore, I do not understand why they are not differentiated in hypercomplex variables. Or is it appropriate to take any x?

May I pop in another minor question? In Z=1x+iy+ jz+kct, I know c is the characteristic speed of the medium, what if we were dealing with water, or peroleum? Does this mean that that number was in mind when deriving this transformation? I am solving the Burgers equation, which does not require scaling the time, so why should I scale it here?
Thanks a million

 

[mathwonk]

sorry for not answering. we may not know what hypercomplex numbers are. in an old algebra book i have from over 50 years ago, it says hypercomplex systems are called "nowadays" algebras.

so hypercomplex systems language has gone out of usage some 50-70 years ago.

i apologize, but we would answer if we understood the question better.

best regards.

 

[jim mcnamara]

Hypercomplex numbers are defined as extensions of the complex numbers. They are constructed in algebra: quaternions are the only ones I ever encountered - that I remember. Hamilton discovered them. Maybe that's what the OP means...

Quaternions can be represented by a 2X2 matrix of complex numbers.

I don't quite get the question, either, but that's normal. :)

 

[mathwonk]

those are called algebras "nowadays", i.e. for the last 70-80 years, they are generally not necessarily commutative rings containing another commutative ring or field in their center, such as the quaternions are an algebra over the reals.

there are others, such as the cayley numbers, but these are the only algebraic ones i think. herstein has a little section on this.