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Chapter4_22 - PHY4604 R D Field Rotations in 3 Dimensional...

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PHY4604 R. D. Field Department of Physics Chapter4_22.doc University of Florida Rotations in 3 Dimensional Space (2) Rotations About the x-Axis, y-Axis, and z-Axis: It is easy to show that = ) cos( ) sin( 0 ) sin( ) cos( 0 0 0 1 ) ( x x x x x x R φ = ) cos( 0 ) sin( 0 1 0 ) sin( 0 ) cos( ) ( y y y y y y R = 1 0 0 0 ) cos( ) sin( 0 ) sin( ) cos( ) ( z z z z z z R Arbitrary Rotation the 3-Space: The most general rotation in 3-space can be written as R( φ x , φ y , φ z ) = R x ( φ x )R y ( φ y )R z ( φ y ) ( i.e. three real parameters ). Infinitesimal Generators of O(3): Look at the infinitesimal transformation z z z z z z z z z z I I R δφ + = = = 1 0 0 0 1 0 1 1 0 0 0 ) cos( ) sin( 0 ) sin( ) cos( ) ( where I used 1 ) cos( 0  → ε and  → 0 ) sin( . Thus, r I I r R r z z z z r r r ) ( ) ( ' + = = and ) ( ) ( ) ( z z z z z z r I r r r r r = + . Hence r I d r d z z r r = and thus r e r z z I r r = ' where = 0 0 0 0 0 1 0 1 0 z I and = = 1 0 0 0 )
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