M402-Chapter2

M402-Chapter2 - i Chapter II: General Coordinate...

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i Figure 1 . The Cartesian coordinate system for 3- dimensional Euclidian space. Chapter II: General Coordinate Transformations Before beginning this chapter, please note the Cartesian coordinate system belowand the definitions of the angles θ and φ in the spherical coordinate system. In the spherical coordinate system, (r, θ , φ ) we shall use: and in the cylindrical coordinate system ( ρ , φ ,z): x = r sin θ cos φ y = r sin θ sin φ z = r cos Θ x = ρ cos φ y = ρ sin φ ρ 2 = x 2 + y 2 .
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II-1 dx i ± ± f ± q 1 dq 1 ² ± ± 2 2 ² ± ± 3 3 ± ² j ± x ± also ± ± ± ² k ± ± ± ± using chain rule ± ± ± ² A . ± ± ± ± ± ± ± ² B . ± ± Chapter II: General Coordinate Transformations Consider two coordinate systems in 3-dimensional Euclidian space: 1. a Cartesian system where a point is specified by (x 1 , x 2 , x 3 ) ± (x,y,z) 2. a general "q i " system where a point is specified by (q 1 , q 2 , q 3 ). Each point (x 1 , x 2 , x 3 ) corresponds to a unique set of real numbers (q 1 , q 2 , q 3 ). Further, each x i is a function of the q j , f i (q 1 ,q 2 ,q 3 ), and each q j = h j (x 1 ,x 2 ,x 3 ) where all first partial derivatives of f i and h j exist. Using only the chain rule for differentiation, the following equations can be obtained: where we have used the notation x i (q 1 ,q 2 ,q 3 ) ± f i (q 1 ,q 2 ,q 3 ) and q j (x 1 ,x 2 ,x 3 ) ± h j (x 1 ,x 2 ,x 3 ) . Note that the differential, dx i , "transforms with" [ ² x i / ² q j ] and the partial derivative "transforms with" [ ² q j / ² x i ]. So, using the summation notation (repeated indices ==> summation): Note the placement of the indices.
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II-2 A i . j ± ² x ² q ± contravariant transformation matrix . B . ± ² ² ± covariant transformation matrix dx dy dz ± 1 2 3 ± sin ± cos ² rcos ± cos ² ² rsin ± sin ² sin ± sin ² ± sin ² ± cos ² cos ± ² ± 0 dr d ± ² ± ² ± dq 1 2 3 In general, ± x i / ± q j ² ± q j / ± x i . This means that dx i and ± / ± x i transform differently under the coordinate transformation. The two transformations have been named contravariant and covariant , respectively. Example: Transformation from Cartesian to spherical coordinates : x = rsin θ cos φ ± x 1 r = [x² + y² + z²] ½ ± q 1 y = rsin θ sin φ ± x 2 θ = cos -1 (z/r) ± q 2 z = rcos θ ± x 3 φ = tan -1 (y/x) ± q 3 The differentials are given by: dx = sin θ cos φ dr + rcos θ cos φ d θ + -rsin θ sin φ d φ dy = sin θ sin φ dr + rcos θ sin φ d θ + rsin θ cos φ d φ dz = cos θ dr + -rsin θ d θ + 0 · d φ Thus,
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II-3 dq k ' M q k M x j dx j ' [ A & 1 ] k . j dx j M M q n ' M x m M q n M M x m ' [ B & 1 ] n . m M M x m In the above equation A 1 1 = sin θ cos φ and A 3 2 = -rsin θ , etc. The elements of the contravariant transformation matrix are obtained from the expression for the differentials of dx, dy and dz. One can also show that: dr d d rr r dx dy dz θ θϕ θ ϕ θ θ ϕ θ ϕθ =− sin cos sin sin cos ( c o sc o s) / ( c o ss i n) / s i n/ sin / ( sin ) 0 And using the chain rule one finds: Note that M y/ M θ = rcos θ sin φ is not simply related to M
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M402-Chapter2 - i Chapter II: General Coordinate...

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