The curl of the gradient of "any" scalar field formula_11 is always zero:

One way to establish this identity (and most of the others listed in this article) is to use three-dimensional Cartesian coordinates. According to the article on "curl",

where the right hand side is a determinant, and i, j, k are unit vectors pointing in the positive axes directions, and "∂" = "∂ / ∂ x " "etc". For example, the "x"-component of the above equation is:

where the left-hand side evaluates as zero assuming the order of differentiation is immaterial.

Vector cross product
Product rule for the gradient
किसी अदिश क्षेत्र का ग्रेडिएंट (Gradient of a scalar field)
Curl of the gradient