Engineering Mathematics-I

Matrices, Related matrices, Complex matrices (Hermitian and skew-Hermitian matrices, Unitary matrix), Rank of a matrix, Gauss-Jordan method, Normal form of a matrix, Linear dependence and independence of vectors, Consistency of linear system of equations, Solution of linear system of equations, Characteristic equation, Eigen values, Eigen vectors, Properties of eigen values, Cayley-Hamilton theorem and its applications, Reduction to diagonal form, Quadratic form and their reduction to canonical form.

Review of Limits, Continuity and Differentiability, Mean Value Theorem, Partial Differentiation and its geometrical interpretation, Homogeneous functions, Euler’s theorem and its extension, Total differentials, Composite function, Jacobian, Taylor’s and Maclaurin’s infinite series, Indeterminate forms, Errors and increments, Maxima and minima of functions of two variables, Method of undetermined multipliers. Curve tracing.

Double Integrals (Cartesian and Polar), Change of Order of Integration, Change of Variables, Applications of Double Integrals. Triple integrals, Change of Variables, Applications of Triple Integrals. Beta and Gamma functions.

Differentiation of vectors, Curves in space, Velocity and acceleration, Relative velocity and acceleration, Scalar and vector point functions, Vector Operator ‘Del’ - Del Applied to Scalar Point Functions (Gradient) and its Geometrical Interpretation - Directional Derivative, Del Applied to Vector Point Function (Divergence and Curl) and their Physical Interpretation, Del Applied Twice to Point Function, Del Applied to Products of Point Functions. Integration of Vector, Tangential Line Integral, Normal Surface Integral, Volume integrals, Theorems of Green, Stokes and Gauss (without proofs) and their verifications and applications, Irrotational Fields, Solenoidal Fields.

Euler’s formula, Dirichlet’s Conditions, Functions Having Points of Discontinuity, Change of interval, Expansion of odd and even periodic functions, Half-range series, Typical wave-forms, Parseval’s formula, Practical harmonic analysis