The Curl of a Gradient is Always Zero
Formulas
The curl of the gradient of a scalar function eq1 is always eq2.
eq1
Proof
In Cartesian coordinates, the gradient of eq1 is as follows.
eq2
When we compute the curl of eq4, it is as follows.
eq3
Since the result is eq6 for all components, regardless of eq1, the rotation of the gradient is always eq2.
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