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Rotations about the x, y, zaxes of the Bloch sphere are represented by the rotation operator gates Quantum logic gates are represented by unitary matrices A gate which acts on n {\displaystyle n} qubits is represented by a 2 n × 2 n {\displaystyle 2^{n}\times 2^{n}} unitary matrix, and the set of all such gates with the group operation ofZ 2 0 (12r3 − 3r5)dr = 32π (b) F(x,y,z) = (x 2 sin(yz))i (y − xe−z)j z k;If z 2 2z 2 = 0 then both x y2 2x 2 = 0 and 2xy 2y = 0 We begin with the equation from the imaginary part 2xy 2y = 0 2y(x 1) = 0 2y = 0 or x 1 = 0 y = 0 or x = 1 If y = 0, then x2 22x 2 = 0 which has no real solutions (the discriminant b 4ac = ( 2)2 4(1)(2) = 4) Since x is assumed to be real, no solutions result from y = 0 The Divergence Theorem Page 2 X 2 y 2 z 2 2xyz 1