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Two Null Identities

∇ x (∇V) ≡ 0

The curl of the gradient of any scalar field is identically zero.

If a vector field is curl-free ( irrotational vector field / conservative vector field ), then it can be expressed as the gradient of a scalar field.

If ∇ x E = 0, then we can define a scalar field V such that E = -∇V.

The divergence of the curl of and vector field is identically zero.
∇ ∙ (∇ x A) = 0

If a vector field is divergenceless ( solenoidal ), then it can be expressed as the curl of another vector field.

If ∇ ∙ B = 0, then we can define a vector field A such that B = ∇ x A.
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