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 Last modified: 05:42 PM - 29 August, 2022

1. Find a parametrization \(\gamma(t)\) for a line segment joining two given points \((x_1. y_1)\) and \((x_2. y_2)\). Find \(\dot{\gamma}(t)\).

2. What does the parametrization trace out \(\gamma(t) = (\frac{2t}{1+t^2}, \frac{1-t^2}{1+t^2})\)?

3. Show that the parametrization \(\gamma(t) := (t^2 - 1, t(t^2 - 1))\) is not injective, i.e. there are two distinct real numbers \(t_1\) and \(t_2\) so that \(\gamma(t_1) = \gamma(t_2)\). Can you deduce the shape¹ of this curve? Can you express the set of points defined by this curve as the zero set² of some function \(f(x,y)\)?

4. For \(\mathbf{v} : (\alpha,\beta) \to \mathbf{R}^2\) and \(\mathbf{w} : (\alpha,\beta) \to \mathbf{R}^2\), show that \((\mathbf{v}(t). \mathbf{w}(t))' = \mathbf{v}'(t). \mathbf{w}(t) + \mathbf{v}(t). \mathbf{w}'(t)\).

5. If \(\mathbf{n} : (\alpha,\beta) \to \mathbf{R}^2\) is such that \(||\mathbf{n}(t)||\) is constant, then prove that \(\dot{\mathbf{n}}(t)\) is either 0 or perpendicular to \(\mathbf{n}(t)\).

6. if we denote, \[s_\alpha(t) := \int_{t_\alpha}^t| | \dot{\gamma}(u)| | \mathrm{d}u\] \[s_\beta(t) := \int_{t_\beta}^t| | \dot{\gamma}(u)| | \mathrm{d}u\] prove that \(s_\beta(t) - s_\alpha(t)\) is a constant (assume that \(t_\alpha < t_\beta\)).

7. If \(\gamma : (\alpha, \beta) \to \mathbb{R}^2\) is a smooth and regular parametrization, then show that \(| |\dot{\gamma}(t)| | : (\alpha, \beta) \to \mathbb{R}\) is smooth.

8. For the parametrization \(\gamma : (-\pi/2, \pi/2) \to \mathbb{R}^2\) given by \(\gamma(t) = (5\cos(t), 5\sin(t))\),

   1. Find the arc-length function \(s(t)\) (starting at, say, 0)

   2. Find a reparametrization map \(\phi\) so that \(\gamma(\phi(t))\) is a unit-speed parametrization.