082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

26.4.1 Physicalinterpretationof ∇ · v
26.4 The divergence of a vector field 857
If the vector field v represents a fluid velocity field, then, loosely speaking, the divergence
of v evaluated at a point represents the rate at which fluid is flowing away from
or towards that point. If fluid is flowing away from a point then either the fluid density
must be decreasing there or there must be some source providing a supply of new
fluid.
If the divergence of a flow is zero at all points then outflow from any point must be
matchedbyanequalinflowtobalancethis.Suchavectorfieldissaidtobesolenoidal.
Example26.6 Show thatthe vector field
v=xsinyi+ysinxj−z(sinx+siny)k
issolenoidal.
Solution We have
v x
=xsiny so that
Also,
v y
=ysinx so that
Finally,
v z
= −z(sinx +siny)
Therefore,
∂v x
∂x =siny
∂v y
∂y =sinx
sothat
∂v z
∂z
= −(sinx +siny)
∇·v=siny+sinx−(sinx+siny)=0
and hencevissolenoidal.
Engineeringapplication26.2
ElectricfluxandGauss’slaw
WesawinEngineeringapplication7.5thatelectricchargesproduceanelectricfield,
E, around them which can be visualized by drawing lines of force. Suppose we surroundaregioncontainingchargeswithasurfaceS.Ifasmallportionofthissurface,
δS,ischosenwecandrawthefieldlineswhichpassthroughthisportionasshownin
Figure 26.1.
The flux of E through δS is a measure of the number of lines of force passing
through δS. Gauss’s law states that the total flux out of any closed surfaceSis proportional
to the total charge enclosed. It is possible to show that this law can be
expressed mathematically as
∇·E= ρ ε 0
➔