Determination of nth derivative of standard functions-illustrative examples*.Leibnitz’s theorem (without proof) and problems.Rolle’s Theorem – Geometrical interpretation. Lagrange’s and Cauchy’smean value theorems. Taylor’s and Maclaurin’s series expansions of functionof one variable (without proof).
Indeterminate forms – L’Hospital’s rule (without proof), Polar curves: Anglebetween polar curves, Pedal equation for polar curves. Derivative of arclength – concept and formulae without proof. Radius of curvature - Cartesian,parametric, polar and pedal forms.
Partial differentiation: Partial derivatives, total derivative and chain rule,Jacobians-direct evaluation.Taylor’s expansion of a function of two variables-illustrative examples*.Maxima and Minima for function of two variables. Applications – Errors andapproximations.
Scalar and vector point functions – Gradient, Divergence, Curl, Laplacian,Solenoidal and Irrotational vectors.Vector Identities: div (øA), Curl (øA) Curl (grad ø ) div (CurlA) div (A x B )& Curl (Curl A) .Orthogonal Curvilinear Coordinates – Definition, unit vectors, scale factors,orthogonality of Cylindrical and Spherical Systems. Expression for Gradient,Divergence, Curl, Laplacian in an orthogonal system and also in Cartesian,Cylindrical and Spherical System as particular cases – No problems
Differentiation under the integral sign – simple problems with constantlimits.Reductionformulaefortheintegralsofn x , cos n x, mn xand evaluation of these integrals withsins inx cosstandard limits - Problems.Tracing of curves in Cartesian, Parametric and polar forms – illustrativeexamples*. Applications – Area, Perimeter, surface area and volume.222Computation of these in respect of the curves – (i) Astroid: x 3 + y 3 = a 3(ii)Cycloid:r = a (1 + cos θ )
Solution of first order and first degree equations: Recapitulation of themethod of separation of variables with illustrative examples*. Homogeneous,Exact, Linear equations and reducible to these forms. Applications -orthogonal trajectories.
Recapitulation of Matrix theory. Elementary transformations, Reduction ofthe given matrix to echelon and normal forms, Rank of a matrix, consistencyof a system of linear equations and solution. Solution of a system of linearhomogeneous equations (trivial and non-trivial solutions). Solution of asystem of non-homogeneous equations by Gauss elimination and Gauss –Jordan methods.
Linear transformations, Eigen values and eigen vectors of a square matrix,Similarity of matrices, Reduction to diagonal form, Quadratic forms,Reduction of quadratic form into canonical form, Nature of quadratic forms