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Unformatted text preview: example: Prove that A ( B C ) = C ( A B ) solution: Use the definition, Eq. 18, ( B C ) i = j,k ijk B j C k (22) then A ( B C ) i = i j,k ijk A i B j C k = i,j,k ijk A i B j C k (23) Similarly, the right-hand side of the equation is C ( A B ) = i,j,k ijk C i A j B k (24) By definition, interchanging two adjacent signs of ijk changes its sign: C ( A B ) = i,j,k- jik C i A j B k = i,j,k jki A j B k C i (25) Since the i, j, k are dummy indices, they can be renamed, and LHS=RHS. 1.4 Vector calculus If a vector is a function of a scalar variable (for example, A ( t )), its derivative is defined by the limit: d A dt = lim t A ( t + t )- A ( t ) t (26) Where the subtraction in the numerator is now a vector subtraction. This allows us to work by components: for example d A dt = dA x dt , dA y dt , dA z dt = A (27) The rules of differentiation are similar to those for ordinary scalar functions. Let A ( t ) and B ( t ) be vector functions of the scalar variable...
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- Spring '10