Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR CALCULUS10.21 Paraboloidal coordinates u, v, φ are defined in terms of Cartesian coordinates byx = uv cos φ, y = uv sin φ, z = 1 2 (u2 − v 2 ).Identify the coordinate surfaces in the u, v, φ system. Verify that each coordinatesurface (u = constant, say) intersects every coordinate surface on which one ofthe other two coordinates (v, say) is constant. Show further that the system ofcoordinates is an orthogonal one and determine its scale factors. Prove that theu-component of ∇×a is given by(1 aφ(u 2 + v 2 ) 1/2 v + ∂a )φ− 1 ∂a v∂v uv ∂φ .10.22 Non-orthogonal curvilinear coordinates are difficult to work with and should beavoided if at all possible, but the following example is provided to illustrate thecontent of section 10.10.In a new coordinate system for the region of space in which the Cartesiancoordinate z satisfies z ≥ 0, the position of a point r is given by (α 1 ,α 2 ,R), whereα 1 and α 2 are respectively the cosines of the angles made by r with the x- andycoordinateaxes of a Cartesian system and R = |r|. The ranges are −1 ≤ α i ≤ 1,0 ≤ R