# class2 - example Prove that A Â B Ã— C = C Â A Ã— B...

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Unformatted text preview: example: Prove that A Â· ( B Ã— C ) = C Â· ( A Ã— B ) solution: Use the definition, Eq. 18, ( B Ã— C ) i = j,k Îµ ijk B j C k (22) then A Â· ( B Ã— C ) i = i j,k Îµ ijk A i B j C k = i,j,k Îµ ijk A i B j C k (23) Similarly, the right-hand side of the equation is C Â· ( A Ã— B ) = i,j,k Îµ ijk C i A j B k (24) By definition, interchanging two adjacent signs of Îµ ijk changes itâ€™s sign: C Â· ( A Ã— B ) = i,j,k- Îµ jik C i A j B k = i,j,k Îµ 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 Î” t â†’ A ( t + Î” t )- A ( t ) Î” 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 = Ë™ 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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class2 - example Prove that A Â B Ã— C = C Â A Ã— B...

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