434 CHAPTER 4. MAPPINGS AND TRANSFORMATIONS and ∂m ∂θ = ∂f ∂x ∂x ∂θ + ∂f ∂y ∂y ∂θ = p − y x 2 P ( − r sin θ ) + ± 1 x ² ( r cos θ ) = r 2 sin 2 θ r 2 cos 2 θ + r sin θ r cos θ = tan 2 θ + 1 = sec 2 θ. While this may seem a long-winded way to go about performing the diFerentiation (why not just diFerentiate tan θ ?), this point of view has some very useful theoretical consequences, which we shall see later. Similarly, change-of-coordinate transformations in three variables can be handled via their Jacobians. The transformation Cyl : R 3 → R 3 going from cylindrical to rectangular coordinates is just Pol together with keeping z unchanged: Cyl ( r,θ,z ) = r cos θ r sin θ z
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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.