Unformatted text preview: 304 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION so differentiating x ′ ( τ ) = △ s ∂x ∂s ( s ( τ ) ,t ( τ )) + △ t ∂x ∂t ( s ( τ ) ,t ( τ )) = parenleftBig radicalbig △ s 2 + △ t 2 parenrightBig v j ( s,t,θ ) which has absolute value at least ( K/ 2) radicalbig △ s 2 + △ t 2 , and in particular is nonzero. Thus the value of the x coordinate is a strictly monotonic function of τ along the curve −→ p ( s ( τ ) ,t ( τ )) joining −→ p ( s 1 ,t 1 ) to −→ p ( s 2 ,t 2 ), and hence the points are distinct. The parametrization of the sphere (Equation ( 3.25 )) shows that the conclusion of Proposition 3.5.7 breaks down if the parametrization is not regular: when φ = 0 we have −→ p ( φ,θ ) = (0 , , 1) independent of θ ; in fact, the curves corresponding to fixing φ at a value slightly above zero are circles of constant latitude around the North Pole, while the curves corresponding to fixing θ are great circles, all going through this pole. This is reminiscent of the breakdown of polarthrough this pole....
View Full Document
This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
- Fall '08