Lecture 1 Terminology from linear algebra: the scalar product of X, Y ∈ R2 is �X, Y � = X1 Y1 + X2 Y2 . The length of a vector is �X� = �X, X�1/2 . The rotation by any angle α is the linear transformation of R2 with matrix � � cos(α) − sin(α) A= . sin(α) cos(α) � � In particular, J = 01 −01 is anticlockwise rotation by 90 degrees. We write det(X, Y ) for the determinant of the matrix with column vectors X, Y ∈ R2 . Equivalently, det(X, Y ) = �JX, Y � or �X, Y � = det(X, JY ). Finally, suppose that X ∈ R2 is any vector, and Y ∈ R2 is a vector of length one. Then X = �Y, X�Y + det(Y, X)JY. Terminology from calculus: a map is called smooth if it is inﬁnitely diﬀer entiable. Lemma 1.1. Let I ⊂ R be an interval, and f : I → R2 a smooth map such that �f (t)� = 1 for all t. Then f � (t) = det(f (t), f � (t))Jf (t). Definition 1.2. A regular curve is a smooth map c : I → R2 , where I ⊂ R is an interval, satisfying c� (t) �= 0 for all t. The curvature of c at t is κ(t) = det(c� (t), c�� (t)) . �c� (t)�3 In physics terminology, if distance in R2 is measured in meters m, and time on I in seconds s, then κ is of type 1/m. For instance, a circle of radius R has curvature 1/R if it is parametrized in an anticlockwise way, and −1/R if it is parametrized in a clockwise way. Proposition 1.3 (Frenet equation of motion). For a regular curve c, � � � d c (t) c� (t) � = �c (t)�κ(t)J = κ(t) Jc� (t). dt �c� (t)� �c� (t)� Corollary 1.4. If κ(t) = 0 for all t, then c(I) ⊂ R2 is part of a straight line. Corollary 1.5. Suppose that κ(t) = 1/R is a nonzero constant. Then � c + RJ �cc� � is constant, and therefore c is part of a circle of radius |R|.