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Unformatted text preview: 304 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION so differentiating x ( ) = s x s ( s ( ) ,t ( )) + t x t ( s ( ) ,t ( )) = parenleftBig radicalbig s 2 + t 2 parenrightBig v j ( s,t, ) 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 along the curve p ( s ( ) ,t ( )) 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 = 0 we have p ( , ) = (0 , , 1) independent of ; in fact, the curves corresponding to fixing at a value slightly above zero are circles of constant latitude around the North Pole, while the curves corresponding to fixing are great circles, all going through this pole. This is reminiscent of the breakdown of polarthrough this pole....
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- Fall '08