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Consider a vector field <math>\vec{F}=f_1\vec{i}+f_2\vec{j}+f_3\vec{k}</math>, which we interpret as representing the velocity of a fluid occupying space, or a very thin layer of flowing fluid on a plane. Suppose we wished to find out if there are any sources of fluid (like a faucet), or drains - how might we go about this?
Consider a vector field <math>\vec{F}=f_1\vec{i}+f_2\vec{j}+f_3\vec{k}</math>, which we interpret as representing the velocity of a fluid occupying space, or a very thin layer of flowing fluid on a plane. Suppose we wished to find out if there are any sources of fluid (like a faucet), or drains - how might we go about this?
Well, consider some specific point, <math>(x_0,y_0,z_0)=\vec{p}</math>. If we surround <math>\vec{p}</math> with a sphere, call it ''S'', then obviously
<math>\oiint_S \vec{F} dS </math>
will contain some contribution from whatever "flow" is coming out of, or going into, <math>\vec{p}</math>. It will, of course, also contains contributions from the various other points in its interior, but, if we let the radius ''r'' of the sphere ''S'' go to 0, then this integral will ONLY measure the flow at <math>\vec{p}</math>. Hence we define the '''divergence''' of a vector field <math>\vec{F}</math> as
<math>\div \vec{F} = \lim_{r\to 0} \oiint_S \vec{F} dS </math>
==Computation==
==Computation==