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Re: Energy



Regarding Brian W.'s fearsome self-proclaimed mangling:

But to strike fear in the heart of any reasonably rigorous educator,
as I believe David to be, I will now proceed to mangle the pristine
description he gave, while retaining some unrecognizable fragment
of the sentiment, possibly so as to ridicule it?

I start with a vector of mass flow rate spatially extended along a
riparian boundary. I specify a local mass density at some point in
an infinity of locations in the flux. The quotient represents a
volumetric flow rate which would ordinarily be expressed in terms of
volume per time.

Actually, the quotient of j_m/[rho] (mass current density vector
over the mass density) would be expressed as a displaced volume *per
unit area* per unit time directed along the flow direction--which is
*equivalent* to the flow velocity. The total volume of a displaced
parcel that is displaced across a tiny area patch of a given surface
(perpendicular to the flow) in a given tiny time interval is the
product of the area of the perpendicular patch and the distance the
parcel moves along the flow in that amount of time. Dividing this
volume by the perpendicular patch area gives just the directed
distance moved (i.e. displacement) along the flow by the parcel of
'mass fluid' in this amount of time. Dividing this displacement by
the tiny time interval over which it occurred gives the velocity of
the parcel. Thus the vector field directed along the flow of the
displaced volume per unit area per unit time *is* the velocity field
for the flowing continuous mass distribution.

When the velocity field is investigated it will be observed that its
magnitude never exceeds c anywhere for any physical flow of a
continuous distribution of an extensive locally conserved quantity.

I take a particle of volume so small that it has
no spatial extension, and proclaim that when this particle passes a
bounding surface, no time has elapsed.

OK, (except for the minor part about the volume per *unit area* per
unit time). Also, we of course need to properly take limits to
find the instantaneous velocity, i.e.
(zero displacement)/(zero time interval).

Did I get the general idea? :-)

More or less, apparently.

David Bowman
David_Bowman@georgetowncollege.edu