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 Last modified: 04:01 PM - 17 October, 2022

1. Prove that \(\ddot{\gamma}(t).\mathbf{\hat{n}}(\gamma(t)) = -\dot{\gamma}(t).\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{\hat{n}}(\gamma(t))\). Here \(\mathbb{\hat{n}}\) denotes the normal to a surface. (Hint: You have done similar things many times before. Which rule helps you?)

2. Use the chain rule to show that \(\frac{\mathrm{d}}{\mathrm{d}t}|_{t=t_0}\mathbf{\hat{n}}(\gamma(t)) = \frac{\mathrm{d}}{\mathrm{d}t}|_{t=t_0}\mathbf{\hat{n}}(\delta(t))\) if \(\gamma(t_0) = \delta(t_0)\) and \(\dot{\gamma}(t_0) = \dot{\delta}(t_0)\). In other words, the derivative is the same for curves which pass through the same point \(\gamma(t_0)\) and have the same velocity vectors. (Hint: Look at everything from the point of view of the coordinate patch and apply chain rule).

3. Can you see how the previous two exercises show that for a unit speed parametrization \(\gamma\), the quantity \(\ddot{\gamma}(t).\mathbf{\hat{n}}(\gamma(t))\) depends only on the point and direction.

4. Prove that \(\mathbf{\hat{n}}(\gamma(t))\) is perpendicular to \(\mathbf{T}(t)\), where \(\mathbf{T}(t)\) is the unit tangent of \(\gamma\) at \(t\).

5. Consider a parametrization, \(\gamma(t)\) and let \(\mathbf{N}(t)\) denote its unit normal at \(t\). Prove that \(\mathbf{N}(t) = \mathbf{\hat{n}}(\gamma(t))\) if and only if \(\kappa_g(t)=0\)

6. Consider the sphere of radius 1. Give it a surface patch and use that to compute the normal curvature of a unit speed parametrization of any curve on the sphere at any given point.

7. Consider any plane. Give it a surface patch and use that to compute the normal curvature of a unit speed parametrization of any curve on that plane at any given point.