Mathematics II

syllabus for Mathematics II in the second semester of an engineering program:

Unit I: Calculus of Several Variables

  • Functions of Several Variables: Limits, continuity, and partial derivatives
  • Homogeneous Functions: Euler’s theorem for functions of two and three variables
  • Total Derivatives: Chain rule, implicit functions
  • Extrema of Functions: Lagrange multipliers

Unit II: Multiple Integrals

  • Double Integrals: Cartesian and polar coordinates, change of order of integration
  • Triple Integrals: Cartesian, cylindrical, and spherical coordinates
  • Applications: Area, volume, center of mass, and moment of inertia

Unit III: Vector Calculus

  • Vector Algebra: Dot product, cross product, and their applications
  • Vector Differentiation: Gradient, divergence, curl, and their physical interpretations
  • Vector Integration: Line integrals, surface integrals, Green’s theorem, Stokes’ theorem, and Gauss’ divergence theorem

Unit IV: Complex Analysis

  • Complex Numbers: Algebra, polar form, and exponential form
  • Analytic Functions: Cauchy-Riemann equations, harmonic functions
  • Complex Integration: Line integrals in the complex plane, Cauchy’s integral theorem and formula
  • Series: Taylor and Laurent series, residue theorem, and applications

Unit V: Differential Equations

  • First-Order Differential Equations: Exact, linear, and Bernoulli equations
  • Higher-Order Linear Differential Equations: Homogeneous and non-homogeneous equations, method of undetermined coefficients, variation of parameters
  • Series Solutions: Power series solutions, Frobenius method

Unit VI: Laplace Transforms

  • Definition and Properties: Linearity, shifting theorems, and initial/final value theorems
  • Inverse Laplace Transform: Convolution theorem, partial fraction decomposition
  • Applications: Solving ordinary differential equations and systems of differential equations

Unit VII: Fourier Series and Transforms

  • Fourier Series: Periodic functions, orthogonality, Fourier coefficients
  • Fourier Transforms: Properties, applications to boundary value problems

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