Partial differntials with hypercomplex numbers

In summary, the conversation discusses the use of hypercomplex numbers in solving partial differential equations and their connection to algebraic structures such as quaternions. The question is raised about the differentiation of constants in hypercomplex variables and the role of characteristic speeds in different media. The conversation also mentions the historical usage of the term "hypercomplex systems" and the current terminology of "algebras."
  • #1
mrcuteblackie
1
0
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
 
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  • #2
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.
 
  • #3
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. :)
 
  • #4
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.
 

1. What are partial differentials?

Partial differentials are a mathematical concept used to calculate the rate of change of a function with respect to one of its variables while holding the other variables constant. They are used in multivariable calculus and are represented by the symbol ∂.

2. What are hypercomplex numbers?

Hypercomplex numbers are a generalization of complex numbers that have more than two imaginary units. They are also known as higher-dimensional numbers and include quaternions, octonions, and other algebras.

3. How are partial differentials calculated with hypercomplex numbers?

Partial differentials with hypercomplex numbers are calculated in a similar way as with real numbers. The only difference is that the imaginary units in hypercomplex numbers follow different algebraic rules, which can make the calculations more complex.

4. What are some real-world applications of partial differentials with hypercomplex numbers?

Partial differentials with hypercomplex numbers are used in many fields, such as physics, engineering, and economics. They are particularly useful in studying fluid dynamics, electromagnetism, and optimization problems.

5. Are there any limitations to using partial differentials with hypercomplex numbers?

Yes, there are some limitations to using partial differentials with hypercomplex numbers. One major limitation is that not all of the properties of real numbers apply to hypercomplex numbers, so some calculations may not be possible. Additionally, the use of hypercomplex numbers may not always provide a more accurate or useful solution compared to using real numbers.

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