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Lists in Mathematica are represented by a pair of braced brackets which contain the elements of the list separated by commas. To save the list for future use, we set the list equal to the variable listf by listf = 8Cos@xD, Sin@xD< 8Cos@xD, Sin@xD< 24 1. 1) is executed by the sequence AppendToAlistf, 2 i listfP2T y listfP1T IntegrateAj cccccccc z z , xEE êê Simplify j cccccccccccccccc k listfP1T { 8Cos@xD, Sin@xD, x Cos@xD + Sin@xD< in which we append the result from an integration of the iteration formula to listf by means of the function AppendTo[].

18) We demonstrate this relation for the 2ä2 rotation matrices. 2 Mathematical Tools R=J cosHfL sinHfL N -sinHfL cosHfL ij cosHfL sinHfL yz z j k -sinHfL cosHfL { The inverse of this matrix is R-1 êê Simplify ij cosHfL -sinHfL yz z j k sinHfL cosHfL { and the transpose is RT cosHfL -sinHfL y zz jij k sinHfL cosHfL { obviously both matrices are equivalent. This result in two dimensions can be generalized to higher dimensions and to the general representation of l. l T == IdentityMatrix@3D 88O211 + O212 + O213 , O11 O21 + O12 O22 + O13 O23 , O11 O31 + O12 O32 + O13 O33 <, 8O11 O21 + O12 O22 + O13 O23 , O221 + O222 + O223 , O21 O31 + O22 O32 + O23 O33 <, 8O11 O31 + O12 O32 + O13 O33 , O21 O31 + O22 O32 + O23 O33 , O231 + O232 + O233 << == 881, 0, 0<, 80, 1, 0<, 80, 0, 1<< We observe that if the matrix l is orthogonal, the off-diagonal elements have to vanish and the diagonal elements are identical to 1.

Let us first consider the case in which the coordinate axes are rotated counterclockwise through an angle of 90° about the x3 -axis. In such a rotation, xè 1 = x2 , xè 2 = -x1 , and xè 3 = x3 . The only nonvanishing cosines are cosHxè 1 , x2 L = 1 = l12 , cosHxè 2 , x1 L = -1 = l21 , cosHxè 3 , x3 L = 1 = l33 . 21) Thus the l matrix for this case is i 0 1 0z y j j 1 0 0 z z Ox3 = j j z j z j z k 0 0 1{ ij 0 1 0 yz jj z jj -1 0 0 zzz jj zz k 0 0 1{ The multiplication of this transformation matrix with vectors along the three coordinate axes shows us how the coordinate axes are changed.

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A Three Dimensional Model For Direct Laser Metal Powder Deposition And Rapid Prototyping by Labudovic, Kovalevic

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