Engineering Calculus Notes 196

Engineering Calculus Notes 196 - function of the form...

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184 CHAPTER 2. CURVES so the tangent line is the x -axis, while when t = 1, dx dt = 0 dy dt = 3 2 so the tangent line is the y -axis. Note in particular that near t = 1 we have y as a function of x while near t = 1 the opposite is true. In a case like this, one refers to diFerent branches of the curve through the origin. We shall not pursue this issue further in the text, but Exercise 14 investigates a further complication, which shows that Lemma 2.4.2 does not have an analogue for regular curves that are not parametrized in a one-to-one manner. In particular, it shows that we cannot automatically assume that two regular vector-valued functions with the same set of points as images describe the same “curve”; we need them to be reparametrizations of each other as well. Regular Curves in Space The theory we have outlined for planar curves applies as well to curves in space. A regular parametrization of a curve in space is a C 1 vector-valued
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Unformatted text preview: function of the form −→ p ( t ) = ( x ( t ) ,y ( t ) ,z ( t )) = ( x ( t )) −→ ı + ( y ( t )) −→ + ( z ( t )) −→ k with non-vanishing velocity −→ v ( t ) := V p ′ ( t ) = ( x ′ ( t ) ,y ′ ( t ) ,z ′ ( t )) = ( x ′ ( t )) −→ ı + ( y ′ ( t )) −→ + ( z ′ ( t )) −→ k n = −→ (or equivalently, non-zero speed) ( x ′ ( t )) 2 + ( y ′ ( t )) 2 + ( z ′ ( t )) 2 n = 0 . We can no longer talk about such a curve as the graph of a function, but we can get a kind of analogue of the second statement in Proposition 2.4.7 which can play a similar role: Remark 2.4.8. If −→ p ( t ) is a regular vector-valued function with values in R 3 , then locally its projections onto two of the three coordinate planes are graphs: more precisely, for each parameter value t = t at least one of the...
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