MATH

21MAT11

Module - 1

 Differential Calculus – 1
Polar curves, angle between the radius vector and the tangent, angle between two curves. Pedal
equations. Curvature and Radius of curvature – Cartesian, Parametric, Polar and Pedal forms.
Problems.
Self-study: Center and circle of curvature, evolutes and involutes.

Module - 2

Differential Calculus – 2
Taylor’s and Maclaurin’s series expansion for one variable (Statement only) – problems.
Indeterminate forms-L’Hospital’s rule. Partial differentiation, total derivative-differentiation of
composite functions. Jacobian and problems. Maxima and minima for a function of two variables.
Problems.
Self-study: Euler’s Theorem and problems. Method of Lagrange undetermined multipliers with
single constraint.

Module - 3

 Ordinary Differential Equations (ODE’s) of first order
Linear and Bernoulli’s differential equations. Exact and reducible to exact differential equations.
Applications of ODE’s-Orthogonal trajectories, Newton’s law of cooling.
Nonlinear differential equations: Introduction to general and singular solutions; Solvable for p only;
Clairaut’s equations, reducible to Clairaut’s equations. Problems.
Self-Study: Applications of ODE’s: L-R circuits. Solvable for x and y.

Module - 4

Ordinary Differential Equations of higher order
Higher-order linear ODE’s with constant coefficients – Inverse differential operator, method of
variation of parameters, Cauchy’s and Legendre homogeneous differential equations. Problems.
Self-Study: Applications to oscillations of a spring and L-C-R circuits

Module - 5

Linear Algebra
Elementary row transformation of a matrix, Rank of a matrix. Consistency and Solution of system of
linear equations; Gauss-elimination method, Gauss-Jordan method and Approximate solution by
Gauss-Seidel method. Eigenvalues and Eigenvectors-Rayleigh’s power method to find the dominant
Eigenvalue and Eigenvector.
Self-Study: Solution of system of equations by Gauss-Jacobi iterative method. Inverse of a square
matrix by Cayley- Hamilton theorem

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