Mathematical Analysis II Syllabus

• Laplace transform (10 hours)

• Differentiating functions of several variables (20 hours)
  • Review on vectors and elements of topology of $\mathbb{R}^n$.
  • Functions of several variables, vector fields.
  • Limits and continuity.
  • Partial and directional derivatives, Jacobian matrix.
  • Differentiability, gradient and tangent plane.
  • Second derivatives, Hessian matrix.
  • Taylor polynomial.
  • Critical points, free extrema.
• Integrating functions of several variables (30 hours)
  • Double and triple integrals, center of mass.
  • Length of a curve and area of a graph.
  • Line and surface integrals (graphs only), circulation and flux of a vector field.
  • Conservative vector fields.
  • Green, Gauss and Stokes theorems.
• Complex analysis and Series (40 hours)
  • Function theory of complex variable: differentiability, Cauchy-Riemann equations.
  • Complex line integrals, Cauchy integral formula.
  • Definition and convergence criteria for numerical series.
  • Power series (real and complex). Taylor series and Laurent series. Residue theorem
  • Fourier series.