(a) Linear transformations of the plane
i. Affine planes and vector spaces
ii. Vector spaces and their affine spaces
iii. Euclidean and affine transformations
iv. Representing linear transformations by matrices
v. Areas and determinants
(b) Eigenvectors and eigenvalues
i. Conformal linear transformations
ii. Eigenvectors and eigenvalues
iii. Markov processes
(c) Linear differential equations in the plane
i. Functions of matrices
ii. Computing the exponential of a matrix
iii. Differential equation and phase portraits
iv. Applications of differential equations
(d) Scalar products
i. The Euclidean scalar product
ii. Quadratic forms and symmetric matrices
iii. Normal modes
iv. Normal modes in higher dimensions
v. Special relativity: The Poincare’ group and the Galilean group
(e) Calculus in the plane
i. The differential calculus and the examples of the chain rule: the Born approximation and Kepler motion
ii. Partial derivatives and differential forms.
iii. The pullback notation
iv. Taylor’s formula
v. Lagrange multiplier
(f) Double integrals
i. Exterior derivative
ii. Two-forms
iii. Pullback and integration for two-forms
iv. Two-forms in three space
v. Green’s theorem in the plane