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=The Divergence=
=The Divergence=
==Definition==
We've discussed partial derivatives, and we've even discussed a way to take a sort of "complete" derivative of multivariable functions (the gradient). But the gradient only works for scalar functions; how about a "complete" kind of derivative for vector fields?
In two dimensions, there is just one, but in three dimensions, there are, for reasons which won't appear until [[Differential Geometry|much later]] in your mathematical education. The first of these is called the divergence, which we investigate now.
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?
=The Curl=
=The Curl=