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Engineering Calculus Notes 447

# Engineering Calculus Notes 447 - 4.2 DIFFERENTIABLE...

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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 θ 0 sin θ r cos θ 0 0 0 1 · sin φ ρ cos φ 0 0 0 1 cos φ ρ sin φ 0 = 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 φ 0 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 φ 0 . (4.6) This can be used to study motion which is most easily expressed in spherical coordinates. For example, suppose
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