What is the Integral of a Volume?

  • #1

Main Question or Discussion Point

We know the relationship of:

speed > acceleration > Jerk > Jounce (or snap)

What is the similar relation ship of:

length > area > volume > ???

My question is: what comes after volume in this pattern?

-4th dimensional volume?

Also how do I refer to this?

-The integral of volume?

Thank you for your consideration,

Brian Jester
 

Answers and Replies

  • #2
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Talking of "length", "area" , "volume" means that we are in the field of geometry, respectively in 1, 2, 3 dimensions spaces. So, extending the concept implies to go to the 4th dimension and the corresponding name is "hypervolume", in fact 4D.hypervolume. Then we have 5-D.hypervolume, ..., n-D.hypervolume.
The formulas corresponding to hyper-sphere, hyper-cone, hyper-sphreical cap in n-dimensions spaces are available in the paper "Le problème de l'hyper-chèvre" :
http://www.scribd.com/JJacquelin/documents
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
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The "integral of a volume" doesn't mean anything. You integrate functions, not geometric quantities. But that just says that "integral" is the wrong word.

In three dimensional geometry there is nothing "beyond" volume. Mathematically, you can, however, talk about n-dimensional spaces for any integer n. In that case, for any [itex]m\le n[/itex], we can talk about the "m-volume" or "m-measure". "Length" is "1-volume" or "1-measure", "area" is "2-volume" or "2-measure", etc.

By the way, "speed > acceleration > Jerk > Jounce (or snap)" are physics terms, not mathematics.
 
  • #4
JJacquelin,

Thank you for the explanation, after revisiting the hypercube, I can conceptualize a hyper volume.

Brian Jester
 

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