Two Null Identities

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 , then we can define a scalar field V such that . The divergence of the curl of and vector field is identically zero. If a vector field is divergenceless ( solenoidal ), then it can be expressed as the curl of another vector field. If , then we can define a vector field such that .