300 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Another function is −→ p ( r,θ ) = ( r cos θ,r sin θ, 0) or x = r cos θ y = r sin θ z = 0 which describes the xy-plane in polar coordinates; the partials are ∂ −→ p ∂r ( r,θ ) = (cos θ ) −→ ı + (sin θ ) −→ ∂ −→ p ∂θ ( r,θ ) = − ( r sin θ ) −→ ı + ( r cos θ ) −→ ; these are independent unless r = 0, so we get a regular parametrization of the xy-plane provided we stay away from the origin. We can similarly parametrize the sphere of radius
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