0.1. What one needs to know?

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What one needs to know?

  1. Multivariable Calculus
  2. Ordinary Differential Equations

Assets: (useful but not required)

  1. Complex Variables,
  2. Elements of (Real) Analysis,
  3. Any courses in Physics, Chemistry etc using PDEs (taken previously or now).

Multivariable Calculus

Differential calculus
  1. Partial Derivatives (first, higher order), differential, gradient, chain rule;
  2. Taylor formula;
  3. Extremums, stationary points, classification of stationart points using second derivatives; Asset: Extremums with constrains.
  4. Familiarity with some notations Section A.2.
Integral calculus
  1. Multidimensional integral, calculations in Cartesian coordinates;
  2. Change of variables, Jacobian, calculation in polar, cylindrical, spherical coordinates;
  3. Path, Line, Surface integrals, calculations;
  4. Green, Gauss, Stokes formulae;
  5. $\nabla u$, $\nabla \times A$, $\nabla \cdot A$, $\Delta u$ where $u$ is a scalar field and $A$ is a vector field. See also Section A.1.

Ordinary Differential Equations

First order equations
  1. Definition, Cauchy problem, existence and uniqueness;
  2. Equations with separating variables, integrable, linear.
Higher order equations
  1. Definition, Cauchy problem, existence and uniqueness;
Linear equations of order $\ge 2$
  1. General theory, Cauchy problem, existence and uniqueness;
  2. Linear homogeneous equations, fundamental system of solutions, Wronskian;
  3. Method of variations of constant parameters.
Linear equations of order $\ge 2$ with constant coefficients
  1. Fundamental system of solutions: simple, multiple, complex roots;
  2. Solutions for equations with quasipolynomial right-hand expressions; method of undetermined coefficients.
Systems
  1. General systems, Cauchy problem, existence and uniqueness;
  2. Linear systems, linear homogeneous systems, fundamental system of solutions, Wronskian;
  3. Method of variations of constant parameters;
  4. Linear systems with constant coefficients.

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