Engineering Calculus Notes 447

Engineering Calculus Notes 447 - 4.2. DIFFERENTIABLE...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.
Ask a homework question - tutors are online