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

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iFigure 1. The Cartesian coordinate system for 3-dimensional Euclidian space.Chapter II: General Coordinate TransformationsBefore beginning this chapter, please note the Cartesian coordinate system belowand the definitionsof the anglesθandφin 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 sinθcosφy = r sinθsinφz = r cosΘx =ρcosφy =ρsinφρ2= x2+ y2.
II-1dxifiq1dq1fiq2dq2fiq3dq3jxiqjdqjalsoxikqkxiqkusing chain ruledxixiqjdqjAi.jdqjxiqjxiqjBi.jqjChapter II: General Coordinate TransformationsConsider two coordinate systems in 3-dimensional Euclidian space:1. a Cartesian system where a point is specified by (x1, x2, x3) (x,y,z)2. a general "qi" system where a point is specified by(q1, q2, q3).Each point (x1, x2, x3) corresponds to a unique set of real numbers (q1, q2, q3).Further, each xiis a function of the qj,fi(q1,q2,q3), and each qj= hj(x1,x2,x3) where all first partial derivatives of fiand hjexist.Using only the chain rulefor differentiation, the following equations can be obtained:where we have used the notation xi(q1,q2,q3) fi(q1,q2,q3) and qj(x1,x2,x3)hj(x1,x2,x3) .Note that the differential, dxi, "transforms with" [ xi/ qj]and the partial derivative "transforms with"[ qj/ xi].So, using the summation notation (repeated indices ==> summation):Note the placement of the indices.
II-2Ai.jxiqjcontravariant transformation matrix.Bi.jqjxicovariant transformation matrixdxdydzdx1dx2dx3sin cosrcoscosrsinsinsin sinrcossinrsincoscosrsin0drdddrdddq1dq2dq3In general,xi/ qjqj/ xi.This means thatdxiand/ xitransform differently under thecoordinate transformation.The two transformations have been namedcontravariantandcovariant,respectively.Example:Transformation from Cartesian to spherical coordinates:x = rsinθcosφx1r = [x² + y² + z²]½q1y = rsinθsinφx2θ= cos-1(z/r)q2z = rcosθx3φ= tan-1(y/x)q3The differentials are given by:dx = sinθcosφdr +rcosθcosφdθ+-rsinθsinφdφdy = sinθsinφdr +rcosθsinφdθ+rsinθcosφdφdz =cosθdr +-rsinθdθ+0 ·dφThus,
II-3dqk'MqkMxjdxj'[A&1]k.jdxjMMqn'MxmMqnMMxm'[B&1]n.mMMxmIn the above equationA11= sinθcosφandA32= -rsinθ,etc.The elements of the contravariant transformationmatrix are obtained from the expression for the differentials of dx, dy and dz.One can also show that:drddrrrrrdxdydzθθϕθϕθθϕθϕθϕθϕθ=sincossinsincos(coscos) /(cossin) /sin/sin/ ( sin )cos/ ( sin )0And using the chain rule one finds:Note thatMy/Mθ= rcosθsinφis not simply related toMθ/My = (cosθsinφ)/r.One can see also from the aboveequations thatA-1=BTandB-1=AT.In the above expressions different summation indices were used foreach of the sums.This is a good habit to adopt.Try to do this in all your derivations.

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Term
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Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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