# B-Course of Mathematics (Paper A & B)

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 and onward organized by University of Sargodha, Sargodha.

• NOTE: attempt two questions from each section.

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. Differential operators. Application to vector analysis. Composition and resolution of co-planar forces. $(\lambda ,\mu )$ Theorem, Lamy’s Theorem, Varignon’s Theorem, Moments, couples and conditions of equilibrium under the action of co-planar forces.

Types of forces, Direction of forces of constraints, Equilibrium of three co-planer forces and related problem. Center of gravity. Symmetry and Center of mass, Center of mass of various bodies. Frictional forces. Laws of friction. Equilibrium of bodies on rough surfaces. Principle of virtual work and related problems.

Kinematics of a particle in Cartesian and polar co-ordinates. Laws of mechanics. Linear and angular velocity. Relative velocity. Rectilinear motion with uniform and variable acceleration. Simple harmonic motion. Projectile motion. Motion along horizontal and vertical circles. Orbital motion. Elliptic orbit under a central force, Polar form of the orbit, Apse and Apsidal Distance, Planetary motion and Keplar’s laws.

Chapter 2 (Vector Analysis)
Chapter 2 Vector in three diamensions
Chapter 2 Scalar and vector products with applications
Chapter 2 Scalar and vector triple products
Chapter 3 (Vector Analysis)
Chapter 3 Differentiation and integration of vector functions
Chapter 4 (Vector Analysis)
Chapter 4 Gradient, divergence and cur, Differential operators
Chapter 4 Application to vector analysis
Chapter 2 (Mechanics)
Chapter 2 Composition and resolution of co-planar forces
Chapter 2 (λ, μ) Theorem, Lamy’s Theorem, Varignon’s Theorem, Moments
Chapter 2 couples and conditions of equilibrium under the action of co-planar forces
Chapter 3 (Mechnics)
Chapter 3 Types of forces
Chapter 3 Direction of forces of constraints
Chapter 3 Equilibrium of three co-planer forces and related problem
Chapter 4 (Mechnics)
Chapter 4 Center of gravity, Symmetry and Center of mass
Chapter 4 Center of mass of various bodies
Chapter 5 (Mechnics)
Chapter 5 Frictional forces, Laws of friction
Chapter 5 Equilibrium of bodies on rough surfaces
Chapter 6 (Mechnics)
Chapter 6 Principle of virtual work and related problems
Chapter 7 (Mechanics)
Chapter 7 Kinematics of a particle in Cartesian and polar co-ordinates
Chapter 7 Laws of mechanics, Linear and angular velocity, Relative velocity
Chapter 8 (Mechanics)
Chapter 8 Rectilinear motion with uniform and variable acceleration
Chapter 8 Simple harmonic motion
Chapter 10 (Mechanics)
Chapter 10 Projectile motion
Chapter 10 Motion along horizontal and vertical circles
Chapter 12 (Mechanics)
Chapter 12 Orbital motion, Elliptic orbit under a central force
Chapter 12 Polar form of the orbit, Apse and Apsidal Distance
Chapter 12 Planetary motion and Keplar’s laws
• NOTE: attempt two questions from each section.

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. Power series solution about an arbitrary point.

Laplace Transformation, solution of ODEs. Error analysis. Solution of non-linear (algebraic and transcendental) equation in one variable using bisection method, false position method, Newton - Raphson method and fixed point method. Difference operators. Interpolation (Newton’s and Lagrange’s methods). Numerical differentiation (at a point of the data). Numerical integration (rectangular, trapezoidal and 1/3 Simpson’s rules).

Co-ordinates in three dimension. Rectangular, cylindrical and spherical co-ordinates. Equations of plane, straight line, sphere, cylinder, cone, ellipsoid, hyperboloid and paraboloid. Longitude and latitudes. Spherical triangle and direction of Qibla. Inner Product Space, Eigen values and Eigen vectors, Dignalization of matrices.

Chapter 9 (Method)
Ex 9.2, 9.3 Classification and formation of DEs
Ex 9.4 to Ex 9.6 Various methods of solutions of first order ODE (linear and non-linear)
Ex 9.8 The Bernoulli’s, Ricatti and Clairaut’s equations
Ex 9.9 Singular solutions
Ex 9.7 Orthogonal trajectories
Chapter 10 (Method)
Ex 10.1, 10.2 Linear DE of higher order (homogeneous and non-homogeneous)
Ex 10.3 Solution by: D-operator and undetermined co-officients Methods
Ex 10.4 Cauchy-Euler equation
Ex 10.5, 10.6 Reduction of order and variation of parameters methods for 2nd order linear DE
Ex 10.7 Power series solution about an arbitrary point
Ex 10.11 Application of first order ODE in problems of decay and growth
Ex 10.11 Population dynamics, Logistic equations
Chapter 10 (Calculus)
Ex 10.1, 10.2
Ex 10.3 Solution of ODEs
Numerical Analysis Error analysis
Numerical Analysis Solution of non-linear (algebraic and transcendental) equation in one variable using bisection method
Numerical Analysis False position method,Newton - Raphson methodFixed point method
Numerical Analysis Difference operators
Numerical Analysis Interpolation (Newton’s and Lagrange’s methods)
Numerical Analysis Numerical differentiation (at a point of the data)
Numerical Analysis Numerical integration (rectangular, trapezoidal and 1/3 Simpson’s rules)
Chapter 8 (Calculas)
Ex 8.1 Co-ordinates in three dimension
Ex 8.7 Rectangular, cylindrical and spherical co-ordinates
Ex 8.4, 8.5, 8.6 Equations of plane
Ex 8.2, 8.3 Straight line
Ex 8.11 Sphere
Ex 8.9, 8.10, 8.12 Cylinder, Cone, Ellipsoid, Hyperboloid and paraboloid
Ex 8.13 Longitude and latitudes
Ex 8.13 Spherical triangle and direction of Qibla
Chapter 7 (Method)
Ex 7.1, 7.2 Inner Product Space
Ex 7.3 Eigen values and Eigen vectors
Ex 7.4 Dignalization of matrices
1. Theory of Differential Equations of Dennis G.Zill. Books Thomson Learning Academic Resource Center. USA.
2. Mathematical Techniques by K.H. Dar, Irfan-ul-Haq and M.A. Jajja. The Carvan Books House. Kachehry Road, Lahore.
3. Mathematics Methods by S.M. Yousaf. Illmi Kitab Khana. Urdu Bazar, Lahore.
4. Numerical Analysis by R.L. Burden and J.D. Faires. PES-Kent Publishing Company. Bostan. USA
5. Operations Research by H.A. Taha. Prentice-hall Inc. Englewood. Cliffs USA.(1996)
6. Mathematical Statistics. By Dr. J.E.Freund. Prentice-hall Inc. Englewood. Cliffs USA.
7. Vector and Tensor Methods, by Chorlton, Ellis Horwood Publishers.
8. Elementary Vector Analysis. By Dr. Munawar Hussain. S.M. Hafeez. M.A. Saeed and Ch. Bashir Ahmed. The Caravan Book House, Kachhry Road , Lahore.
9. A Text Book by Dynamics by Chorlton, Van Nostrand Company Ltd. London.
10. Mechanics by O.K. Ghori. West Pakistan Publishing Company, Lahore.
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