$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ ## What one needs to know? > 1. [Multivariable Calculus](#calculus2) > 2. [Ordinary Differential Equations](#ode) **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](../SA.2.html). #####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](../SA.1.html). ####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. [$\Leftarrow$](./S0.html) [$\Uparrow$](../contents.html) [$\Rightarrow$](../Chapter1/S1.1.html)