Engineering Calculus Notes 189

Engineering Calculus Notes 189 - 177 2.4. REGULAR CURVES...

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2.4. REGULAR CURVES 177 Now, consider the vector-valued function −→ q ( θ ) = (cos θ ) −→ ı + (sin θ ) −→  , 0 < θ < π giving another parametrization of the semicircle, in terms of polar coordinates. This vector-valued function is also one-to-one and C 1 , with nonvanishing velocity vector at θ = θ 0 −→ v −→ q ( θ 0 ) = V q ( θ 0 ) = ( sin θ 0 ) −→ ı + (cos θ 0 ) −→  . Comparing the two parametrizations, we see that each point of the semicircle corresponds to a unique value of θ 0 as well as a unique value of x 0 ; these are related by x 0 = cos θ 0 . In other words, the function −→ q ( θ ) can be expressed as the composition of −→ p ( x ) with the change-of-variables function x = cos θ. Furthermore, substituting cos θ 0 for x 0 (and noting that for 0 < θ 0 < π , sin θ 0 > 0, so r 1 x 2 0 = sin θ 0 ) we see that at the point P = −→ p ( x 0 ) = p x 0 , R 1 x 2 0 P = −→
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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.

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