we wish to evince in vector terms 'except' for the minus sign. Since the differential operators in each term act either over one vector (say ’s) or the other (’s), the contribution to each term must be These results can be rigorously proven to be correct through evaluation of the vector components. Therefore, the RHS can be written in vector form as Putting together these two results, a result analogous to Green's theorem for scalar fields is obtained, Theorem for vector fields: The curl of a cross product can be written as Green's vector identity can then be rewritten as Since the divergence of a curl is zero, the third term vanishes to yield Green's vector identity: With a similar procedure, the Laplacian of the dot product can be expressed in terms of the Laplacians of the factors As a corollary, the awkward terms can now be written in terms of a divergence by comparison with the vector Green equation, This result can be verified by expanding the divergence of a scalar times a vector on the RHS. See also Green's function Kirchhoff integral theorem Lagrange's identity (boundary value problem) References External links Green's Identities at Wolfram MathWorld Vector calculus Mathematical identities