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Engineering Calculus Notes 316

Engineering Calculus Notes 316 - 304 CHAPTER 3 REAL-VALUED...

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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 , 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 polar coordinates at the origin. A point at which a C 1 function −→ p : R 2 R 3 has dependent partials (including the possibility that at least one partial is the
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