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→Applications of the Divergence and Curl
=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.'''
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.'''
If we return now the the fluid understanding of a vector field, imagine putting a paddlewheel in a particular spot. The fluid might make the paddlewheel spin if we put in in along one axis, and might not if it goes in along another axis. The axis along which the wheel will spin ''fastest'' is the direction of <math>\nabla \times \vec{F}</math>, and the magnitude of <math>\nabla \times \vec{F}</math> tells us how fast the paddlewheel will spin.
So, if a vector field satisfies <math>\nabla \times \vec{F}=\vec{0}</math>, it means it's not spinning around any axis, anywhere. It makes sense to call such a field '''irrotational'''.
{{MVCalc}}
{{MVCalc}}