Calc3.6

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 much later in your mathematical education. The first of these is called the divergence, which we investigate now.

Consider a vector field ${\vec {F}}=f_{1}{\vec {i}}+f_{2}{\vec {j}}+f_{3}{\vec {k}}$, 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?

geometric definition

Computation

This is an unwieldy definition, so we frequently use the equivalent statement

$div{\vec {F}}={\frac {\partial f_{1}}{\partial x}}+{\frac {\partial f_{2}}{\partial y}}+{\frac {\partial f_{3}}{\partial z}}$

or in two dimensions, just ${\frac {\partial f_{1}}{\partial x}}+{\frac {\partial f_{2}}{\partial y}}$. Note that if we were to define a "vector" $\nabla ={\vec {i}}\partial _{x}+{\vec {j}}\partial _{y}+{\vec {k}}\partial _{z}$, we would have $div{\vec {F}}=\nabla \cdot {\vec {F}}$. Note that this isn't technically a vector, since the component are not number but differential operators, and the kind of "multiplication" we're doing isn't really multiplication, since you can't multiply an operator by a function; instead we're differentiating the components. Nevertheless, it's a handy notational convenience, and it actually does point to a deeper truth we won't begin to address in this course, so we'll continue to use $\nabla \cdot {\vec {F}}$ for the divergence.