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303Appendix

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

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19.1. APPENDIX 19.1 Appendix 19.1 Alternate Notations unit vectors: ˆ x, ˆ y, ˆ z , or ˆ i, ˆ j, ˆ k , or ˆ a x , ˆ a y , ˆ a z , or a x , a y , a z ( sometimes bold ) coordinate positions: ( x, y, z ), or ( r, φ, z ), or ( r, θ, φ ) ( 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 ˆ x , d y ˆ x , ~ d s , or d z ˆ z , ~ d S , ~ d a , ~ d A 19.2 Right-Handed Orthogonal Coordinate Systems Cartesian: ( x, y, z ); ~ A = A x ˆ x + A y ˆ y + A z ˆ z position vector: ~ r( x, y, z ) = x ˆx + y ˆ r + z ˆ z differential examples: ~ d = d x ˆ x , ~ d S = d x d y ˆ z , d V = d x d y d z Cylindrical: ( r, φ, z ); ~ A = A r ˆ r + A φ ˆ φ + A z ˆ z with ˆ r and ˆ φ parallel to the x - y plane. position vector: ~ r( r, φ, z ) = r ˆ r( φ ) + z ˆ z differential examples: d = d x , or ~ d = d r ˆ r , ~ d = r d θ ˆ θ , ~ d = d z ˆ z ~ d S = r d θ d z , d V = r d r d θ d z Spherical: ( r, θ, φ ); ~ A = A r ˆ r + A θ ˆ θ + A φ ˆ φ position vector: ~ r( r, θ, φ ) = r ˆ r( θ, φ ) differential examples: d = dr , d = r d θ , d = r sin θ d φ , ~ d = dr ˆ r , ~ d = r d θ ˆ θ , ~ d = r sin θ d φ ˆ φ , ~ d S = r 2 sin θ d θ d φ d V = (d r ) · ( r d θ ) · ( r sin θ d φ ) = r 2 sin θ d r d θ d φ “right-hand rule”: x y z x , r φ z r , and r θ φ r Vector equations (e.g., Maxwell’s Equations) give exactly the same solution in any coordinate system; hence, use the coordinate system that takes greatest advantage of any symmetries in the problems’s configuration. 19.3 Vector Operations and Operators Dot product: ~ A · ~ B = AB cos 6 AB , ˆ x · ˆ x = 1, ˆ r · ˆ r = 1, ˆ φ · ˆ φ = 1, ˆ x · ˆ y = 0, ˆ r · ˆ φ = 0 Cross product: ~ A × ~ B = ˆn AB sin 6 AB (ˆn to plane defined by ~ A and ~ B), ˆ x × ˆ x = 0, ˆ r × ˆ r = 0, ˆ φ × ˆ φ = 0, ˆ x × ˆ y = ˆ z , ˆ y × ˆ z = ˆ x , ˆ y × ˆ x = - ˆ z , ˆ r × ˆ φ = ˆ z , ˆ r × ˆ θ = ˆ φ See section 20.5 for the Gradient, Divergence, and Curl in different coordinate systems.

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19.2 19.4 Coordinate Transformations Table 19.1: Coordinate Tranformations - Scalar Quantities Cartesian Cylindrical Spherical x = r cos φ = r sin θ cos φ y = r sin φ = r sin θ sin φ z = z = r cos θ p x 2 + y 2 = r = r sin
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303Appendix - 19.1 APPENDIX 19.1 Appendix 19.1 Alternate...

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