304CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATIONso differentiatingx′(τ) =△s∂x∂s(s(τ),t(τ)) +△t∂x∂t(s(τ),t(τ))=parenleftBigradicalbig△s2+△t2parenrightBigvj(s,t,θ)which has absolute value at least (K/2)radicalbig△s2+△t2, and in particular isnonzero. Thus the value of thexcoordinate is a strictly monotonicfunction ofτalong the curve−→p(s(τ),t(τ)) joining−→p(s1,t1) to−→p(s2,t2),and hence the points are distinct.The parametrization of the sphere (Equation (3.25)) shows that theconclusion of Proposition3.5.7breaks down if the parametrization is notregular: whenφ= 0 we have−→p(φ,θ) = (0,0,1)independent ofθ; in fact, the curves corresponding to fixingφat a valueslightly above zero are circles of constant latitude around the North Pole,while the curves corresponding to fixingθare great circles, all goingthrough this pole. This is reminiscent of the breakdown of polarcoordinates at the origin. A point at which aC1function−→p:R2→R3hasdependent partials (including the possibility that at least one partial is the
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North Pole, Polar coordinate system, Coordinate systems