Micromechanics of composites/Proof 6
Curl of the gradient of a vector - 1
editLet be a vector field. Show that
Proof:
For a second order tensor field , we can define the curl as
where is an arbitrary constant vector. Substituting into the definition, we have
Since is constant, we may write
where is a scalar. Hence,
Since the curl of the gradient of a scalar field is zero (recall potential theory), we have
Hence,
The arbitrary nature of gives us