Changes

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

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

<math>\nabla \cdot \vec{F} = \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} + \frac{\partial f_3}{\partial z}</math>

<math>div \vec{F} = \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} + \frac{\partial f_3}{\partial z}</math>

or in two dimensions, just

or in two dimensions, just <math>\frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y}</math>.  Note that if we were to define a "vector" <math>\nabla = \vec{i}\partial_x +\vec{j}\partial_y + \vec{k}\partial_z </math>, we would have <math>div \vec{F} = \nabla \cdot \vec{F} </math>. 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 <math>\nabla \cdot \vec{F} </math> for the divergence.

 

<math>\nabla \cdot \vec{F} = \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y}</math>.

=The Curl=

=The Curl=