Determination of nth order derivatives of Standard functions - Problems.Leibnitz’s theorem (without proof) - problems.Polar Curves - angle between the radius vector and tangent, angle betweentwo curves, Pedal equation for polar curves. Derivative of arc length -Cartesian, Parametric and Polar forms (without proof) - problems. Curvatureand Radius of Curvature – Cartesian, Parametric, Polar and Pedal formsand problems.
Taylor’s and Maclaurin’s theorems for function of one variable(statementonly)- problems. Evaluation of Indeterminate forms.Partial derivatives – Definition and simple problems, Euler’s theorem –problems, total derivatives, partial differentiation of composite functions,Jacobians-definition and problems, extreme values of functions of twovariables.
Derivative of vector valued functions, Velocity, Acceleration and related problems, Scalar and Vector point functions, Gradient, Divergence, Curl,Solenoidal and Irrotational vector fields. Vector identities - div (fA), curl(fA), curl (grad f), div (curl A).Differentiation under integral sign using Leibnitz rule with constant andvariable limits.Curve Tracing - General rules to trace Cartesian, polar and parametriccurves.
Reduction formulae (m and n are positive integers),evaluation of these integrals with standard limits (0 to p/2) and problems.Differential Equations :Solution of first order and first degree differential equations – Exact,reducible to exact and Bernoulli’s differential equations.Applications – orthogonal trajectories, Newton’s law of cooling, flow ofelectricity, laws of decay and growth.
Rank of a matrix by elementary transformations, solution of system of linearequations - Gauss-elimination method, Gauss-Seidel method and L-Udecomposition method.Linear transformation, diagonalisation of a square matrix, Quadratic forms,reduction to Canonical form by orthogonal transformation, Rayleigh’s powermethod to find the largest Eigen value and the corresponding Eigen vector.