303Appendix - 19.1. APPENDIX 19.1 Appendix 19.1 Alternate...

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Unformatted text preview: 19.1. APPENDIX 19.1 Appendix 19.1 Alternate Notations unit vectors: x, y, z , or i, j, k , or a x , a y , a z , or a x , a y , a z ( sometimes bold ) coordinate positions: ( x,y,z ), or ( r,,z ), or ( r,, ) ( never bold ) full vectors: E,E , ~ E, E differential lengths: d , d x , d y , d z differential areas: ~ d S = d x d y~ z , ~ d a , ~ d A , dA differential volume: d V ( never a vector) differential vectors: ~ d , ~ d x , ~ d y , ~ d z , or d x x , d y x , ~ d s , or d z z , ~ d S , ~ d a , ~ d A 19.2 Right-Handed Orthogonal Coordinate Systems Cartesian: ( x,y,z ); ~ A = A x x + A y y + A z z position vector: ~ r( x,y,z ) = x x + y r + z z differential examples: ~ d = d x x , ~ d S = d x d y z , d V = d x d y d z Cylindrical: ( r,,z ); ~ A = A r r + A + A z z with r and parallel to the x- y plane. position vector: ~ r( r,,z ) = r r( ) + z z differential examples: d = d x , or ~ d = d r r , ~ d = r d , ~ d = d z z ~ d S = r d d z , d V = r d r d d z Spherical: ( r,, ); ~ A = A r r + A + A position vector: ~ r( r,, ) = r r( , ) differential examples: d = dr , d = r d , d = r sin d , ~ d = dr r , ~ d = r d , ~ d = r sin d , ~ d S = r 2 sin d d d V = (d r ) ( r d ) ( r sin d ) = r 2 sin d r d d right-hand rule: x y z x , r z r , and r r Vector equations (e.g., Maxwells Equations) give exactly the same solution in any coordinate system; hence, use the coordinate system that takes greatest advantage of any symmetries in the problemss configuration. 19.3 Vector Operations and Operators Dot product: ~ A ~ B = AB cos 6 AB , x x = 1, r r = 1, = 1, x y = 0, r = 0 Cross product: ~ A ~ B = n AB sin 6 AB (n to plane defined by ~ A and ~ B), x x = 0, r r = 0, = 0, x y = z , y z = x , y x =- z , r = z , r = See section 20.5 for the Gradient, Divergence, and Curl in different coordinateSee section 20....
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This note was uploaded on 01/16/2010 for the course ECE 3030 at Cornell University (Engineering School).

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303Appendix - 19.1. APPENDIX 19.1 Appendix 19.1 Alternate...

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