This subject is consists of two papers of 100 marks each. One is called “Paper A” and other is called “Paper B”. This page is updated on February 15, 2015. This syllabus is for 1st Annual 2015 ans onward organized by University of Sargodha, Sargodha.
Theory of limit and continuity. Solution of inequalities. Derivatives and its application to business, economics and physics etc. Differentials. Related rates. Newton-Raphson formula. Higher order derivatives. Leibnitz’s theorem. Limits and continuity of functions of two variables. Partial differentiation and its geometrical meaning for functions of two variables. Euler’s theorem. Increments and differentials. Chain Rule. General theorems (without proofs) and indeterminate forms. L’ Hospital rule of functions. Increasing and decreasing functions.
Translation and rotation of axes. Second degree equation with reference to conic section. Properties of conics. Polar equations of conics. Tangents and normals. Parametric representation of curves. Pedal Equations. Asymptotes. Extrema and its application. Singular points. Curvature. Evolutes and envelopes.
Antiderivatives and indefinite integrals. Methods of integration. Definite integral as limit of sum. Fundamental theorem. Properties. Improper integrals. Reduction formulas. Double and Triple integral (Simple cases). Area between curves. Length of arc. Intrinsic equations. Numerical integration (rectangular, trapezoidal and 1/3 Simpson’s rules). Co-ordinates in three dimension. Rectangular, cylindrical and spherical co-ordinates. Equations of plan, straight line, sphere, cylinder, cone, ellipsoid, hyperboloid and paraboloid. Longitude and latitudes. Spherical triangle and direction of Qibla.
Vectors in three-dimensions. Scalar and vector products with applications. Scalar and vector triple products. Differentiation and integration of vector functions. Gradient, divergence and curl. Sequences. Bounded Sequences. Convergence and divergence of sequences. nth-term test, comparison test, ratio test, root test and integral test for convergence and divergence of infinite series. Convergence and divergence of alternating series. Complex numbers and their properties. De moivre’s theorem and its applications. Circular, logarithmic and hyperbolic functions. Separation into real and imaginary parts.
Algebra of matrices. Determinants with properties. Co-factors, minors, adjoint and inverse of a matrix. Elementary row and column operations. Echelon form and rank of matrix. Solution of the system of linear equations(Homogeneous and non-homogeneous) by use of matrices. Network flow problems. Vector spaces and sub spaces. Linearly dependent and independent vectors. Bases and dimension. Linear transformation. Matrix of linear transformation (relevant theorems of bases and linear transformation with out proofs).
Basic concepts of differential equations. Classification and formation of DEs. Various methods of solutions of first order ODE (linear and non-linear). The Bernoulli’s, Ricatti and Clairaut’s equations. Singular solutions. Orthogonal trajectories. Application of first order ODE in problems of decay and growth, population dynamics, logistic equations. Linear DE of higher order (homogeneous and non-homogeneous). Solution by: D-operator and undetermined co-officients Methods. Reduction of order and variation of parameters methods for 2nd order linear DE. Cauchy-Euler equation.
Note: A student has to take B-course of Mathematics (both I & II- papers) as an additional course to be eligible to do M.Sc. mathematics as a regular or a private candidate.