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appendix_B

# appendix_B - Appendix B Curvilinear coordinate systems q1...

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Appendix B: Curvilinear coordinate systems r ( q 1 , q 2 ) e 1 e 2 q 1 increasing q 2 increasing Many problems can be approached more simply by choosing a coordinate system that fits a given geometry. Instead of writing the position vector r as a function of Cartesian coordinates ( x, y, z ), r is now written as function of three new coordinates: r ( q 1 , q 2 , q 3 ). The coordinate lines are swept out by varying one of the coordinates, keeping the other two constant. We will deal only with the by far most important case of orthogonal coordinate systems, in which the coordinate lines always intersect one another at right angles. Evidently, r ∂q i is a vector which points in the direction of the i-th coordinate line, see the figure. If each of these vectors are normalized to unity, we obtain the local basis system: ˆ q i = 1 h i r ∂q i , h i vextendsingle vextendsingle vextendsingle vextendsingle r ∂q i vextendsingle vextendsingle vextendsingle vextendsingle . (42) The quantities h i ( q 1 , q 2 , q 3 ) are called scale factors or metric coefficients . The fact that

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appendix_B - Appendix B Curvilinear coordinate systems q1...

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