0.1. What one needs to know?

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## What one needs to know?

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.