Changes

Of course, this is useless if we don't know what it measures.  Let's investigate:

Of course, this is useless if we don't know what it measures.  Let's investigate:

[[Image:Curl1.jpg|200px|right|thumb|Here, we see space with the positive ''x''-axis poking out of the screen towards us.  Note that the difference in values and direction of the two partial derivatives is cause by the fact that the direction of the lines coming out of their common basepoint is rotating around in the neighborhood of that point.]]

[[Image:Curl1.jpg|200px|right|thumb|Here, we see space with the positive ''x''-axis poking out of the screen towards us.  Note that the difference in values and direction of the two partial derivatives is cause by the fact that the direction of the lines coming out of their common basepoint is rotating around in the neighborhood of that point.]]

The first component, <math>\left( \partial_y f_3 - \partial_z f_2 \right)</math>, gives us the extent of the vector <math>\nabla \times \vec{F}</math> in the ''x''-axis direction.  The expressions gives us, basically, how fast the fluid moving "up" along the ''z''-axis "tips" in the direction of the ''y''-axis, minus how much the fluid moving in the direction of the ''y'' axis tips in the direction of the ''z'' axis.

The first component, <math>\left( \partial_y f_3 - \partial_z f_2 \right)</math>, gives us the extent of the vector <math>\nabla \times \vec{F}</math> in the ''x''-axis direction.  The expressions gives us, basically, how fast the fluid moving "up" along the ''z''-axis "tips" in the direction of the ''y''-axis, minus how much the fluid moving in the direction of the ''y'' axis tips in the direction of the ''z'' axis.  This is just a measurement of how much the field is rotating about the ''x''-axis (see illustration).  By a similar logic, we see the other components will measure how much the field is rotating about the various other axes, and so when we add these all together, the result will be a vector which A) points along the line around which the field is rotating the most, and B) has magnitdue equal to how fast the fluid is rotating.  For this reason, the expression <math>\nabla \times \vec{F}</math> is often called the '''rotation''' or '''curl''' of <math>\vec{F}</math>, and you will sometimes see <math>\mathbf{rot}\vec{F}</math> or <math>\mathbf{curl}\vec{F}</math> used for <math>\nabla \times \vec{F}</math>.

=Applications of the Divergence and Curl=

=Applications of the Divergence and Curl=

{{MVCalc}}

{{MVCalc}}