interpretation of divergence
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From: aegis on 27 Jun 2010 13:55 This is from Rogawski's text: "let's assume that F is the velocity field of a fluid. Then the flux of F through a surface S is the volume of fluid passing through S per unit time. Suppose that S encloses a region W containing a point P(e.g., a ball of small radius). Then div(F) is nearly constant on W with value div(F)(P), and the Divergence Theorem yields the approximation Flux across S = int( int( int_W( div(F) ) ) dV ~= div(F)(P)Vol(W)" How is div(F) nearly constant on W? -- aegis
From: Robert Israel on 27 Jun 2010 14:01
aegis <aegis(a)mad.scientist.com> writes: > This is from Rogawski's text: > > "let's assume that F is the velocity field of a fluid. Then the flux > of F > through a surface S is the volume of fluid passing through S per unit > time. > Suppose that S encloses a region W containing a point P(e.g., a ball > of small > radius). Then div(F) is nearly constant on W with value div(F)(P), > and the > Divergence Theorem yields the approximation > Flux across S = int( int( int_W( div(F) ) ) dV ~= div(F)(P)Vol(W)" > > How is div(F) nearly constant on W? Assuming the partial derivatives of F are all continuous, div(F) will be continuous. If W is small, this continuous function will be nearly constant there. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada |