29 - Lecture 1 January 4, 2012 1 Parametric equations...

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Unformatted text preview: Lecture 1 January 4, 2012 1 Parametric equations Definition 1.1. A function f : A B is a map which for each point in A assigns a single point in B. Example 1.1. y = + 1 - x2 , x [0, 1]. Example 1.2. y = sin(t), x = cos(t),t [0, 2) The above example is two to one, for x y so is not a function. Example 1.3. y = sin(t), x = cos(t), t [0, ]. Example 1.4. x = sin(0.5 sin(t)), y = cos(0.5 sin(t)) t in[0, 2] Example 1.5. x = sin(0.25 + 0.5 sin(t)), y = cos(0.25 + 0.5 sin(t)), t [0, 2] Example 1.6. x = sin(-t) y = cos(-t) t [0, 100] Remark 1.1. The same curve can have many different parameterizations. Remark 1.2. In some cases only one parameterization is useful, in others, overlaping parameterizations can be useful. Example 1.7. (Pogorelov, Differential Geometry, Noordhoff 1954 pg. 19) A circular disc of radius a rolls uniformly without slipping along a straight line with velocity v. Find the equation of the curve which is described by a point M which is fixed to the circular disc. Under what conditions does the curve have singular points? Homework suggestions Attempt the problems without calculations initially use what you know about common functions, turning points and the behavior at to sketch the curves. Then check your work by calculating a few points. Do a final check using graphing software. 1 ...
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