Mathematics for Physics-S1

Mathematics for Physics

Contact: C. Marinoni

 

Spaces And Vector Spaces

  • Space
  • Measurable spaces
  • Topological spaces
  • Vector spaces
  • Representation of vectors and operators in a basis

Curvilinear Coordinates

  • Coordinates systems
  • Euclidean metric associated to a coordinates system
  • Coordinates transformations
  • Bases associated to a coordinates system
  • Charting a space with coordinates
  • Derivatives of geometric vectors
  • Relation between bases
  • Measuring the extension of figures in space

Integral and differential operators

  • Types of fields
  • Gradient of a scalar field
  • Line integrals of vector fields
  • Surface integrals of vector fields
  • Curl of a vector field
  • The Laplacian
  • Remarkable integral identities
  • Potential Theory

Calculus of Variations

  • Local maxima and minima of functions
  • Lagrange multipliers
  • Functionals
  • First Integral of the Euler-Lagrange equation
  • Variation of constrained functionals
  • Functionals with variable end points
  • Maxima and minima of functionals
  • Fermat’s principle
  • Principle of least action
  • Noether’s theorem
  • Variational principle applied to fields

Orthogonal Functions

  • Fundamental function spaces
  • Orthonormal bases for functions
  • Orthogonal polynomials

Ordinary Differential Equations

  • Ordinary differential equations: a quick review
  • Initial Value Problems for second order linear HODE
  • IVPs for second order linear IODE
  • Boundary values problems
  • BVPs for a second order linear IODE: the eigenfunctions’ method