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Prelim 1 will take place in class on Thursday, February 20. The exam will cover material from section 1.1-1.5, 2.1-2.3, 3.1-3.3, 4.1-4.5. Students are also expected to know how the material in sections 4.1-4.5 generalizes to \(d\)-dimensional manifolds in \(\mathbb{R}^n\) - see these notes. This includes how manifolds can be defined as preimages of regular values under differentiable maps.

The exam may cover any of the following:

• The terminology associated to curves: regular, unit speed, reparameterization, arc length, simple, closed, periodic function. Be familiar with the following quanties and their relationships to eachother: curvature \(\kappa\), torsion \(\tau\), signed curvature \(\kappa_s\), turning angle \(\phi\). Know how to calculuate \(\mathbf{t}\), \(\mathbf{n}\), and \(\mathbf{b}\) as well as the Frenet equations: \[\mathbf{\dot t} = \kappa \mathbf{n} \qquad \mathbf{\dot n} = - \kappa \mathbf{t} + \tau \mathbf{b} \qquad \mathbf{\dot b} = - \tau \mathbf{n}.\]
• Know what data is needed to reconstruct plane curves and space curves up to isometries of \(\mathbb{R}^2\) and \(\mathbb{R}^3\).
• Be familiar with the statements of the following theorems and understand how they can be used: Jordan Curve Theorem, Hopf's Umlaufsatz.
• Be familiar with the statements, proofs, and uses of the following theorems: Four Vertex Theorem, Isoperimetric Inequality, Wirtinger's Inequality.
• Know the following definitions: manifold, surface, smooth manifold, smooth surface, chart, atlas, surface patch, homeomorphism, smooth function (between manifolds), tangent space \(T_p M\) of a manifold \(M\) at a point \(p \in M\), tangent bundle \(TM\), orientation of a manifold, orientation of a surface in \(\mathbb{R}^3\).
• You should be familiar with the methods for proving that a subset \(M \subseteq \mathbb{R}^n\) is a (smooth) manifold.
• Know how to calculate curvature and torsion. You will be given the formulas \[\kappa = \frac{|\!|\dot \gamma \times \ddot \gamma|\!|}{|\!|\dot \gamma|\!|^3} \qquad\qquad \tau = \frac{(\dot \gamma \times \ddot \gamma) \cdot {\dddot \gamma}}{|\!|\dot \gamma \times \ddot \gamma|\!|}.\]
• Be familiar with the following proof techniques:
  • Showing a function is constant by showing its derivative is \(0\).
  • If \(f:[a,b] \to [0,\infty)\) is continuous, then \(\int_a^b f(x) \ dx = 0\) if and only if \(f(x) = 0\) for all \(x \in [a,b]\). This was used repeatedly in the proof of the Isoperimetric Inequality and Wirtinger's Inequality.
  • If \(f,g:[a,b] \to \mathbb{R}\) are continuous and \(f \leq g\), then \(\int_a^b f(x)\ dx \leq \int_a^b g(x) \ dx\). This was used, for instance, when proving that straight lines minimize distance.
  • The chain rule and product rule, including for scalar and vector products (for instance in the verification that \(\kappa = |\!|\dot \gamma \times \ddot \gamma|\!|/|\!|\dot \gamma|\!|^3\)).
  • The extreme value theorem and the intermediate value theorem (for instance in the proof of the Four Vertex Theorem).
  • The geometric properties of the dot and cross product. Some examples of this are in the verification of the Frenet equations, the definition of \(\tau\), and the proof of the Four Vertex Theorem.
  • If \(f'(x) > 0\) on \((a,b)\), then \(f\) is increasing on \((a,b)\).