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→The Curl: this will take some explaining i think...
Since we've taken the dot product of our new differential operator <math>\nabla</math> with a vector field, it only makes sense to see what happens when we take the cross product as well.
Since we've taken the dot product of our new differential operator <math>\nabla</math> with a vector field, it only makes sense to see what happens when we take the cross product as well.
<math>\nabla \times \vec{F} = \begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\ \partial_x & \partial_y & \partial_z \\f_1&f_2&f_3\end{vmatrix} = \left( \frac{\partial f_3}{\partial y} - \frac{\partial f_2}{\partial z} \right)\vec{i} - \left(\frac{\partial f_3}{\partial x} - \frac{\partial f_1}{\partial z} \right) \vec{j} + \left( \frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y} \right) \vec{k} </math>
<math>= \left( \frac{\partial f_3}{\partial y} - \frac{\partial f_2}{\partial z} \right)\vec{i} + \left( \frac{\partial f_1}{\partial z} - \frac{\partial f_3}{\partial x} \right) \vec{j} + \left( \frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y} \right) \vec{k}</math>.
Of course, this is useless if we don't know what it measures. Let's investigate:
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.
=Applications of the Divergence and Curl=
=Applications of the Divergence and Curl=
{{MVCalc}}
{{MVCalc}}