4.2. DIFFERENTIABLE MAPPINGS 435 and the transformation Cyl from cylindrical to rectangular coordinates, which we studied above. Then Sph = ( Cyl ) ◦ ( SC ), and its Jacobian is J ( Sph )( ρ,φ,θ ) = J (( Cyl ) ◦ ( SC ))( ρ,φ,θ ) = J ( Cyl )( r,θ,z ) · J ( SC )( ρ,φ,θ ) = cos θ − r sin θ0 sin θ r cos θ000 1 · sin φ ρ cos φ000 1 cos φ − ρ sin φ0 = sin φ cos θ ρ cos φ cos θ − r sin θ sin φ sin θ ρ cos φ sin θ r cos θ cos φ − ρ sin φ0 and substituting r = ρ sin φ , J ( Sph )( ρ,φ,θ ) = sin φ cos θ ρ cos φ cos θ − ρ sin φ sin θ sin φ sin θ ρ cos φ sin θ ρ sin φ cos θ cos φ − ρ sin φ0 . (4.6) This can be used to study motion which is most easily expressed in spherical coordinates. For example, suppose
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.