Mathematics-II (BIT151HS)

Multiple integrals, ordinary differential equations of the first order, linear differential equations with constant/variable coefficients, Fourier series and integrals, functions of a complex variable, and complex series, residues, and poles.

1. Unit 1: Multiple Integrals6 hrs

   • Double integrals and evaluation in Cartesian coordinates
   • Evaluation of double integrals in polar coordinates
   • Change of order of integration in double integrals
   • Change of variables in double integrals (Jacobian of transformation)
   • Triple integrals in Cartesian coordinates
   • Triple integrals in cylindrical and spherical polar coordinates
   • Applications of multiple integrals: areas of plane regions and volumes of solid bodies
   • Physical applications: centers of gravity, moments of inertia, and mass of lamina

2. Unit 2: Differential Equations of the First Order8 hrs

   • Basic concepts, order, degree, and formation of differential equations
   • Equations of first order and first degree: Variables separable method
   • Homogeneous and reducible to homogeneous differential equations
   • Exact differential equations and necessary and sufficient conditions
   • Integrating factors and rules for finding integrating factors
   • Linear differential equations of first order (Integrating Factor method)
   • Bernoulli's equation (equations reducible to linear form)
   • Equations of the first order but higher degree (solvable for p, solvable for y, solvable for x)
   • Clairaut's equation and singular solutions
   • Applications: Orthogonal trajectories, Newton's law of cooling, chemical mixture problems, growth and decay models

3. Unit 3: Linear Differential Equations with Constant and Variable Coefficients7 hrs

   • Linear differential equations of second and higher orders with constant coefficients
   • Linear independence and dependence of solutions, Wronskian determinant
   • Complementary function (CF) for real, distinct, repeated, and complex roots
   • Particular integral (PI) using operator D = d/dx methods for standard forms: e^{ax}, sin(ax), cos(ax), x^m, and e^{ax}*V(x)
   • Method of variation of parameters for non-homogeneous linear differential equations
   • Cauchy-Euler homogeneous linear equations and reducible forms
   • Legendre's linear differential equations
   • Simultaneous linear differential equations with constant coefficients

4. Unit 4: Fourier Series and Integrals10 hrs

   • Periodic functions and Dirichlet's conditions for Fourier expansion
   • Euler's formulae for Fourier coefficients
   • Fourier series of functions with period 2*pi and arbitrary period 2*L
   • Fourier series for even and odd functions (simplification of coefficients)
   • Half-range Fourier sine series and half-range Fourier cosine series
   • Parseval's identity and its application in summing infinite series
   • Complex exponential form of Fourier series
   • Fourier integral theorem and Fourier integral representations
   • Fourier cosine and Fourier sine transforms
   • Complex Fourier transform, inverse transform, and key properties (linearity, shifting, modulation, convolution)

5. Unit 5: Functions of a Complex Variable8 hrs

   • Review of complex numbers, modulus, argument, and polar representation
   • Neighborhoods, open and closed sets, domains in the complex plane
   • Limits, continuity, and differentiability of complex functions
   • Analytic functions (holomorphic functions) and singular points
   • Cauchy-Riemann (C-R) equations in Cartesian coordinates (necessary and sufficient conditions)
   • Cauchy-Riemann equations in polar coordinates
   • Harmonic functions, harmonic conjugates, and orthogonal families of curves
   • Construction of analytic functions: Milne-Thomson method
   • Conformal mapping: Definition, conditions, scale factor, and angle preservation
   • Bilinear transformation (Mobius transformation), invariant points, cross-ratio preservation

6. Unit 6: Complex Series, Residues and Poles6 hrs

   • Sequences and series of complex numbers, absolute and uniform convergence
   • Power series, radius of convergence, and Cauchy-Hadamard theorem
   • Taylor's theorem and expansion of analytic functions in Taylor series
   • Laurent's theorem and Laurent series expansion about singular points
   • Classification of singularities: Isolated singularities, removable singularities, poles, and essential singularities
   • Zeros of analytic functions and their relationship to poles
   • Residues: Definition and formulas for computing residues at simple and multiple poles
   • Cauchy's Residue Theorem and contour integration fundamentals
   • Evaluation of real definite trigonometric integrals of the type integral from 0 to 2*pi of R(cos theta, sin theta) d theta
   • Evaluation of improper real integrals of rational functions from -infinity to +infinity using semicircular contours