Differences

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--- un:vector-field [2024/10/11 05:08] – [3. Carl] asad
+++ un:vector-field [2024/10/11 08:44] (current) – asad
@@ Line 27: / Line 27: @@
 In the image above, the divergence of the vector field (arrow) $e^{-(x^2+y^2)}(1-2x^2)$ mentioned in the first section is shown in color.
-===== - Carl =====
+===== - Curl =====
 Curl refers to the circulation of a vector field, i.e. how much circular motion there is in that field. Anywhere in the two-dimensional field, if the vectors below a circular area have less magnitude than the vectors above, a clockwise rotation will occur, and the curl will be negative in this case. Curl is positive if the rotation is counterclockwise. The mathematical form of the curl is:
-$$ \nabla\times \mathbf{F} = \left(\frac{\partial F_z}{\partial F_y}-\frac{\partial F_y}{\partial F_z}\right) \hat{i} + \ left(\frac{\partial F_x}{\partial F_z}-\frac{\partial F_z}{\partial F_x}\right) \hat{j} + \left(\frac{\partial F_y}{\partial F_x }-\frac{\partial F_x}{\partial F_y}\right) \hat{k} $$
+$$ \nabla\times \mathbf{F} = \left(\frac{\partial F_z}{\partial F_y}-\frac{\partial F_y}{\partial F_z}\right) \hat{i} + \left(\frac{\partial F_x}{\partial F_z}-\frac{\partial F_z}{\partial F_x}\right) \hat{j} + \left(\frac{\partial F_y}{\partial F_x}-\frac{\partial F_x}{\partial F_y}\right) \hat{k} $$
 {{:bn:un:curl.png?nolink|}}
 In the image above, the $z$-component $e^{-(x^2+y^2)}2xy$ of the curl of the vector field (arrow) mentioned in the first section is shown in color. The curl is a vector quantity, but since the field is two-dimensional here, the x and y components of the curl will be zero.