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=Applications of the Divergence and Curl=

=Applications of the Divergence and Curl=

If we imagine a vector field as being a fluid which fills space, and moves around in it, so that the atom of water at point <math>\vec{p}</math> is travelling in the direction of <math>\vec{F}</math> with speed equal to <math>\vec{F}</math>s magnitude, then the divergence <math>\nabla \cdot \vec{F}</math> simply tells us if it is compressing or decompressing.  For this reason, a vector field such that <math( \nabla \cdot \vec{F})(\vec{p})=0 </math> for all points <math>\vec{p}</math> is called '''incompressible.'''

If we imagine a vector field as being a fluid which fills space, and moves around in it, so that the atom of water at point <math>\vec{p}</math> is travelling in the direction of <math>\vec{F}</math> with speed equal to <math>\vec{F}</math>s magnitude, then the divergence <math>\nabla \cdot \vec{F}</math> simply tells us if it is compressing or decompressing.  For this reason, a vector field such that <math>( \nabla \cdot \vec{F})(\vec{p})=0 </math> for all points <math>\vec{p}</math> is called '''incompressible.'''

Alternatively, we may view a vector field as heat flowing through a solid body, like a metal bar.  Places where the divergence is positive would then be places where heat is being created (by a fire, by contact with a hot body, by nuclear decay, etc.), and for those reasons, points <math>\vec{p}</math> where <math>(\nabla \cdot \vec{F})(\vec{p})>0</math> are called '''sources'''.  Similarly, if the divergence is negative, that means heat is leaving, maybe because of refrigeration, contact with a cold body, or an endothermic reaction of some kind.  For that reason, points <math>\vec{p}</math> where <math>(\nabla \cdot \vec{F})(\vec{p})<0</math> are called '''heat sinks''' or more commonly, simply '''sinks.'''

Alternatively, we may view a vector field as heat flowing through a solid body, like a metal bar.  Places where the divergence is positive would then be places where heat is being created (by a fire, by contact with a hot body, by nuclear decay, etc.), and for those reasons, points <math>\vec{p}</math> where <math>(\nabla \cdot \vec{F})(\vec{p})>0</math> are called '''sources'''.  Similarly, if the divergence is negative, that means heat is leaving, maybe because of refrigeration, contact with a cold body, or an endothermic reaction of some kind.  For that reason, points <math>\vec{p}</math> where <math>(\nabla \cdot \vec{F})(\vec{p})<0</math> are called '''heat sinks''' or more commonly, simply '''sinks.'''