On the existence of the derivative of the volume average |
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Authors: | Jiří Mls |
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Institution: | (1) Czech Technical University, Prague, Czechoslovakia |
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Abstract: | A general theorem on the derivative of the volume average is formulated and proved. Conditions for the existence of the derivative are presented and discussed. This is done in order to give a better base to the theory of spatial averaging.Latin Letters
E
3
three-dimensional vector space over the field of real numbers
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K, K(x)
averaging domain
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G, G
w, Gs
open sets in E
3; components of the two-phase system
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C
1(G)
the set of functions 1-times continuously differentiable in G
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W1/2(G)
Sobolev space
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V
volume of the domain K
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f
function defined in G, G
w
- K
infi
sup*
(x), K
infi
sup–
(x)
special parts of K(x)
Greek Letters
boundary of G, G
w, Gs; w-s interface
-
ij
Kronecker delta
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v
unit outward normal of G, G
w
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j
j-dimensional Lebesgue measure
Other
M
closure of a set M in the metric space E
3
- f
phase average of f for the w-phase
- (u, v)
scalar product of u, v in E
3
-
one-sided derivatives |
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Keywords: | Two-phase system volume average derivative of the volume average |
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