syllabus for Mathematics I

syllabus for Mathematics I in the first semester of an engineering program:

Unit I: Matrices

  • Introduction to matrices
  • Elementary row and column transformations
  • Rank of a matrix
  • Linear dependence and independence
  • Consistency of linear systems of equations
  • Characteristic equation
  • Cayley-Hamilton theorem
  • Eigenvalues and eigenvectors
  • Diagonalization
  • Complex and unitary matrices

Unit II: Calculus

  • Differentiation:
    • Functions, limits, and continuity
    • Derivatives and their applications
    • Mean value theorems
    • Taylor and Maclaurin series
  • Integration:
    • Definite and indefinite integrals
    • Techniques of integration
    • Applications of integrals (area, volume, etc.)

Unit III: Differential Equations

  • First-order differential equations
  • Second-order linear differential equations
  • Applications of differential equations in engineering

Unit IV: Vector Calculus

  • Vector algebra
  • Scalar and vector fields
  • Gradient, divergence, and curl
  • Line and surface integrals
  • Green’s, Stokes’, and Gauss’ theorems

Unit V: Complex Numbers

  • Complex number system
  • Algebra of complex numbers
  • Polar form of complex numbers
  • De Moivre’s theorem
  • Complex functions and their derivatives

Unit VI: Sequences and Series

  • Convergence of sequences and series
  • Tests for convergence
  • Power series
  • Fourier series

Unit VII: Probability and Statistics

  • Basic concepts of probability
  • Random variables and probability distributions
  • Statistical measures (mean, variance, standard deviation)
  • Hypothesis testing and confidence intervals

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