B Vector Calculus Identity proof?

Check here:
http://www-pord.ucsd.edu/~rsalmon/chap4.pdf
Equations 1.2 to 1.4

 

Apparently ## \omega ## is defined as ## \omega=\nabla \times u ##. This one is actually a variation of a well-known E&M vector identity (i.e. a vector calculus identity used quite often in E&M coursework): ## \nabla (a \cdot b)=a \cdot \nabla b +b \cdot \nabla a + \ a \times \nabla \times b + \ b \times \nabla \times a ## ## \ ## where ## a=b=u ##. (This one is found on the cover of J.D. Jackson's E&M textbook as well as other E&M textbooks.)