TEST BANK FOR Introduction to Nonlinear Finite Element Analysis By Nam Ho Kim Converted

1.6. Exercises P1.1 Using Cartesian bases, show that (u Ä v) ⋅ (w Ä x) = (v ⋅ w)u Ä x where u, v, w, and x are rank 1 tensor. Solution: Using the Cartesian basis, ( )( ) ( )( ) i i j j k k l l u Ä v w Ä x = u e Ä v e ⋅ w e Ä x e . Since the dot product occurs between adjacent bases, we have ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) i i j j k k l l i j k l j k i l i j k l jk i l i j j l i l j j i i l l u v w x u v w x u v w x u v w x v w u x d Ä ⋅ Ä = ⋅ Ä = Ä ...