0.1. What one needs to know?
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## What one needs to know?

**Must know:**

- Multivariable Calculus
- Ordinary Differential Equations

**Assets:** (useful but not required)

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

#### Multivariable Calculus

##### Differential calculus

- Partial Derivatives (first, higher order), differential, gradient, chain rule.
- Taylor formula.
- Extremums, stationary points, classification of stationary points using second derivatives.

*Asset:* Extremums with constrains.
- Familiarity with some notations Section A.2.

##### Integral calculus

- Multidimensional integral, calculations in Cartesian coordinates.
- Change of variables, Jacobian, calculation in polar, cylindrical, spherical coordinates.
- Path, Line, Surface integrals, calculations.
- Green, Gauss, Stokes formulae.
- $\nabla u$, $\nabla \times A$, $\nabla \cdot A$, $\Delta u$ where $u$ is a scalar field and $A$ is a vector field. See also Section A.1.

#### Ordinary Differential Equations

##### First order equations

- Definition, Cauchy problem, existence and uniqueness.
- Equations with separating variables, integrable, linear.

##### Higher order equations

- Definition, Cauchy problem, existence and uniqueness.

##### Linear equations of order $\ge 2$

- General theory, Cauchy problem, existence and uniqueness.
- Linear homogeneous equations, fundamental system of solutions, Wronskian.
- Method of variations of constant parameters.

##### Linear equations of order $\ge 2$ with constant coefficients

- Fundamental system of solutions: simple, multiple, complex roots.
- Solutions for equations with quasipolynomial right-hand expressions; method of undetermined coefficients.

##### Systems

- General systems, Cauchy problem, existence and uniqueness.
- Linear systems, linear homogeneous systems, fundamental system of solutions, Wronskian.
- Method of variations of constant parameters.
- Linear systems with constant coefficients.

##### Assets:

- ODE with singular points.
- Some special functions.
Boundary value problems.

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