**Engineering Mathematics Syllabus**

**Eng. Maths – M1 Syllabus**

**Unit – I: Ordinary Differential Equations:**

Basic concepts and definitions of 1st order differential equations; Formation of differential equations; solution of

differential equations: variable separable, homogeneous, equations reducible to homogeneous form, exact differential equation, equations reducible to exact form, linear differential equation, equations reducible to linear form (Bernoulli’s equation); orthogonal trajectories, applications of differential equations.

**Unit – II: Linear Differential equations of 2nd and higher-order:**

Second-order linear homogeneous equations with constant coefficients; differential operators; solution of homogeneous equations; Euler-Cauchy equation; linear dependence and independence; Wronskian; Solution of nonhomogeneous equations: general solution, complementary function, particular integral; solution by variation of parameters; undetermined coefficients; higher order linear homogeneous equations; applications.

**Unit – III: Differential Calculus (Two and Three variables) :**

Taylor’s Theorem, Maxima, and Minima, Lagrange’s multipliers

**Unit – IV: Matrices, determinants, linear system of equations:**

Basic concepts of an algebra of matrices; types of matrices; Vector Space, Sub-space, Basis, and dimension, linear the system of equations; consistency of linear systems; the rank of a matrix; Gauss elimination; the inverse of a matrix by Gauss Jordan method; linear dependence and independence, linear transformation; inverse transformation; applications of matrices; determinants; Cramer’s rule.

**Unit – V: Matrix-Eigen value problems:**

Eigenvalues, Eigenvectors, Cayley Hamilton theorem, basis, complex matrices; quadratic form; Hermitian, Skew Hermitian forms; similar matrices; diagonalization of matrices; transformation of forms to principal axis (conic section).

**Eng. Maths – M2 Syllabus**

**Unit I: Laplace Transforms:**

Laplace Transform, Inverse Laplace Transform, Linearity, transform of derivatives and Integrals, Unit Step function, Dirac delta function, Second Shifting theorem, Differentiation and Integration of Transforms, Convolution, Integral Equation, Application to solve differential and integral equations, Systems of differential equations.

**Unit II: Series Solution of Differential Equations:**

Power series; the radius of convergence, power series method, Fresenius method; Special functions: Gamma function, Beta function; Legendre’s and Bessel’s equations; Legendre’s function, Bessel’s function, orthogonal functions; generating functions.

**Unit III: Fourier series, Integrals and Transforms:**

Periodic functions, Even and Odd functions, Fourier series, Half Range Expansion, Fourier Integrals, Fourier sine, and cosine transforms, Fourier Transform

**Unit IV: Vector Differential Calculus:**

Vector and Scalar functions and fields, Derivatives, Gradient of a scalar field, Directional derivative, Divergence of a vector field, Curl of a vector field.

**Unit V: Vector Integral Calculus:**

Line integral, Double Integral, Green’s theorem, Surface Integral, Triple Integral, Divergence Theorem for Gauss, Stroke’s Theorem

**Eng. Maths M3 Syllabus**

**UNIT I: Basic Probability**

Probability spaces, conditional probability, independent events and Bayes theorem. Random variables: Discrete and continuous random variables, Expectation of random variables, Moments, Variance of random variables.

**UNIT II: Probability distributions:**

Binomial, Poisson, evaluation of statistical parameters for these distributions, Poisson approximation to the binomial distribution.

Continuous random variables and their properties, distribution functions and density functions, Normal and Exponential, evaluation of statistical parameters for these distributions.

**UNIT III: Testing of hypothesis:**

Test of significance: Basic of testing of hypothesis. Null and alternate hypothesis, types of errors, level of significance, critical region.

Large sample test for single proportion, difference of proportions, single mean, difference of means, small sample tests: Tests for single mean, difference of means and test for ratio of variances.

**UNIT IV: Complex variables (Differentiation):**

Limit, Continuity and differentiation of complex functions, Analyticity, Cauchy – Riemann equations (without proof), finding hormonic conjugate, elementary analytic functions and their properties.

**UNIT V: Complex variables (Integration):**

Line integral, Cauchy’s theorem, Cauchy’s integral formula, Zero of analytic functions, singularities, Taylor’s series, Laurent’s series, Residues, Cauchy Residue theorem, conformal mappings, Mobius transformations and their properties.