APPLIED Mathematics syllabus B.G students| UOK

                                B.A/B.Sc. (Three Years Degree Course)
                                            In Applied Mathematics
                 B.A/B.SC. (1stYear) W.e.f. 2006
COURSE NO.                    TITLE OF THE COURSE                           MARKS
AMM-01                             Descriptive Mathematics                                     50
AMM-02                             Differential Calculus                                            50
AMM-03                             Complex Trigonometry and
                                             Theory of Equations                                            50DESCRIPTIVE MATHEMATICS
        AMM-01                                                                                        Marks : 50
Unit I
Sets, Relation and Function:
Review of set operations, Algebra of sets, Cartesians product of sets, Functions,
Composition of functions, Binary Relations, Equivalence relation and partitions,
partial order relation and lattices.  Pigeon Hole Principle.
Linear inequations:

Solution of linear Inequation in one variable and its graphical representation, Solution
of system of linear Inequations in one variable, Graphical solutions of linear
Inequation in two variable, Solution of system of linear Inequations in two variables.

Unit II

Quadratic Equations:
Solution of quadratic equation Relation betweens roots and coefficients, nature of
roots, formation of quadratic equations with given roots, symmetric functions of
roots, equations reducible to quadratic forms.
Sequence and Series:
Sequence and examples of finite and infinite sequences, Arithmetic progression
(A.P), Geometric progression (G.P) and Harmonic progression (G.P) ArithmeticGeometric series, Sum to n terms and sum of infinite arithmetic-geometric series,
special series.
                                 Σn,         Σn²,         Σn³

Unit III

Permutations and Combinations:
Fundamental principle of counting and meaning of n, permutation as arrangements,
meaning of C (n,r), P (n,r). Simple applications including circular permutations.
Principle of Mathematical Induction and its simple applications,  Binomial Theorem
for any index.  Applications of binomial theorem for approximations.  Properties of
Binomial Coefficients.

Unit IV
Coordinate Geometry:
Coordinate systems in a plane, Distance formula, Area of a triangle, Conditions for
collinearty for three points, equations of line, angle between two lines, conditions for
parallelism and perpendicularity.  Distance of a point from a line.  Homogeneous
equation of second degree representing pair of straight lines.  Equation of parabola,
Equations of tangents and normals to a parabola, pole and polar pair of tangents from
a point, parametric equations of parabola.
Ellipse and Hyperbola:
Equation of ellipse, Tangents and Normals, pole and polar, parametric equations of
ellipse, Diameters, conjugate diameters and their properties.
Equations of hyperbola, tangents and normals, equation of hyperbola referred to
asymptotes as axes, Rectangular and conjugate diameters and their properties.
Books Recommended:

1) A text book of Algebra, M.L. Sath
2) A text book of Algebra, G.M. Shah
3) Linear programming, G.Hadley, Harosa Pub. 1995
4) Coordinate Geometry of Conic, M.R.Puri
5) Coordinate Geometry, M.L. Kochar
6) Coordinate Geometry, Ram BallahDifferential Calculus

                        AMM-02                                                                                    Marks : 50
Unit I
Limit of a function, Right hand and left hand limits, ε-δ definition of the limit of a
function.  Basic properties of limits.  Infinitesimals; Definition with examples.
Theorems on infinitesimals.  Comparing infinitesimals, Definition with examples and
related theorems.  Principal part of an infinitesimal and related theorems.  Continuity
and basic properties of continuous functions on closed intervals.  If a function is
continuous in a closed interval, then it is bounded therein.  If a function is continuous
in a closed interval [a, b], then it attains its bounds at least once in [a,b].

Unit II
Definition of a derivative, Derivative as a rate of change, Derivative of some standard
functions, General rules of differentiation, Derivative of a function of a  function,
Differentiation of implicit, circular and inverse circular functions, successive
differentiation, some standard results, Leibnitz Theorem and its applications.

Unit III
Tangents and Normal to plane curves, Equation of the tangent, equation of the
Normal, angle of intersection of two curves, Length of the tangent, sub-tangent,
normal and subnormal, polar coordinates, the polar tangent, polar sub-tangent, polar
normal, polar subnormal.  Pedal equation.  Derivative of the arc (Cartesian and polar
coordinates).
Curvature, radius of curvature for Cartesian and polar coordinates, double points,
Asymptotes, Cartesian and polar coordinates, envelopes, tracing of curves (Cartesian
coordinates only).

Unit IV
Rolle’s theorem with proof and its applications.  Lagrange’s Mean value theorem and
Cauchy’s Mean valu7e theorem with their applications.  Taylor’s and Maclaurin’s
theorem with their applications.  Partial differentiation of functions of two and three
variables.  Euler’s theorem on homogeneous functions.

Books Recommended:

1.  Dr.A.Aziz-ul-Auzeen, S.D Chopra and M.L.Kochar, Differential Calculus
     (Thoroughly revised and enlarged New aEdition 2005-06).
2.  Shanti Narayan, Differential Calculus.
3.  Gorakh Prased, Differential Calculus.Complex Trigonometry and Theory of Equations


         AMM-03                                                                                    Marks : 50
Unit I
Complex number, Argand’s diagram, modulus-amplitude form of a complex number,
Review of complex number system, triangle inequality and its generalization.
Equation of circle (Apollonius circle), Geometrical representation of sun, product and
quotient of two complex numbers.  De Moiver’s Theorem for rational index and its
applications.

Unit II
Expansion of Sin nθ, Cos nθ etc. in terms of powers of Sin θ, Cos θ and expansion of
Sinⁿ θ and Cosⁿ θ in terms of multiple angles of Sinⁿ θ and Cosⁿ θ, Inverse of
trigonometric functions, Functions of complex variable. Exponential, circular.
Hyperbolic, Inverse hyperbolic and Logarithmic functions of a complex variable and
their properties.  Summation of trigonometric series, Difference method, C + iS
method.
THEORY OF EQUATIONS

Unit III
General properties of equations, synthetic division, Relation betweens the roots and
the coefficients of an equation, Transformation of equations, Diminishing the roots of
an equation by a given number, Removal of terms of an equation, Formation of
equations whose roots are functions of the roots of a given equation, Equations of
squared differences.

Unit IV
Symmetric functions, Newton’s method of finding the sum of powers of the roots of
an equation.  Cardan’s solution of the cubic, nature of the roots of a cubic.

BOOKS RECOMMENDED:

1.  Dr. A.Aziz-ul-Auzeem and N.A.Rather, Differential Calculus (Edition 2005).
2.  M.R. Puri,-Complex Trigonometry.
3.  Hem Ram,- Pure Mathematics.
4.  M.L.Sad,-Complex Trigonometry.
5.  Samuel Borofsky,-Elementry theory of Equations.
6.  W.S.Burnisde and A. W. Panton,- Theory of Equations.
7.  J.C. Chaturvedi, - Theory of Equations

•                                          B.A/B.Sc. (Three Years Degree Course)

                                                                   In Applied Mathematics
                                                     B.A/B.SC. (2nd Year) W.e.f. 2006



 

                                                Discrete Mathematics
AMM-04                                                                                              Marks: 50
Unit I:
Mathematical Logic:
Symbolic logic, Statement, Logical connectives, Conjunction, Disjunction, Negation,
Conditional, Bi-conditional sentences, tautology and contradiction, Valid argument,
the structure of mathematical systems, logical equivalence and duality, quantifiers,
Grammars.
Boolean Algebra:
Posets, Lattices, Boolean algebra, operations, Axioms for a Boolean algebra,
Conditional, Bi-conditional statements, validity of arguments, Basic Theorems,
Applications of Boolean algebra to switching circuits.

Unit II:
Graph theory:
Basic definitions, walks, paths and cycles, isomorphism, bipartite graphs, digraphs,
binary operations, Eulerian graphs, Euler’s Theorem, Konisberg bridge problem,
Hamiltonian graphs, Diracs Theorem, traveling salesman problem, trees, cut sets and
cut vertices, Kuratowki’s two graphs, planer graphs, Euler’s formula, graph coloring.

Unit III:
Numerical techniques:
Finite differences, Interpolation with equal and unequal intervals, divided differences,
Newton’s formula for unequal intervals, Sheppard’s rule, Lagrange’s Interpolation
formula,  Hermite’s interpolation formula, Guass interpolation formula, Stirling’s
formula, Bessel’s formula.

                   Solution  of algebraic and transcendental equation-bisection method,
Graphical and analytical, method, Regula falsi method, Secant method, Interation method, Newton Raphson method.

Unit IV:
Laplace Transforms:
Important formula, properties of Laplace transforms,  Laplace transforms of derivatives
and integrals, Unit step function, impulse functions, periodic functions, Convolution
Theorem, Laplace transforms of Bessel function, evaluation of integrals, inverse Laplace
transforms, solution of differential equations.

Books Recommended:
      1. E. Sampath Kumar, K.S. Amer, A brief introduction to Mathematical logic.
      2. Joe L. Mott, Abraham Kadel, discrete Mathematics for Computer Scientists and
Mathematicians.
      3. Schuam’s Series, Laplace transforms.
4. M.K. Jain, S.R.K. Iyenger, R.K. Jain, Numerical methods problems and solutions.
      5. Narsingh Deo, Graph Theory with applications to computer sciences and
engineering, PHI.
6. S. Pirzada and A. Dharwadker, Graph Theory, Universities Press(Orient Longman)
2005.
Books Suggested:
      1. N. Rudraih, E. Sampath Kumar, Discrete Mathematical Structures with Applications
to Computer Science.
      2. J.B. Scarborough, Numerical Mathematical Analysis.
      3. F. Harrary, Graph Theory, PHI.
      4. Atkinson, Numerical Analysis, J. Wiley
      5. Schuam Series, Discrete Mathematics.


             INTEGRAL CALCULUS AND DIFFERENTIAL EQUATIONS
PAPER- AMM-05                                                                                   Marks: 50
                                             (INTEGERAL CALCULUS)
Unit I:
Integration, Integration by substitution and by parts, integration of algebraic rational
functions; case of non-repeated linear factors. Case of linear of quadratic non-repeated
factors, Reduction formulae for the integrals of circular functions and for the integrals of
                          Sin
m
x cos
n
x, cos
m
x cosnx, x
m
cosnx
Unit II:
Definite integral as the limit of a sum. Summation of series with the help of definite
integrals, Quaderature, Area of a region bounded by a curve, X-axis (y-axis) and two
ordinates (abscissa), Sectorial areas bounded by a closed curve. Length of plane curves.
Volumes and surfaces of revolution.
                                           

                                            DIFFERENTIAL EQUATIONS
Unit III:
Degree and order of a differential equations, Equations of first order and first degree.
Equations in which the variables are separable. Homogeneous equations, Linear equations
and equations reducible to linear form. Bernoulli’s equations. Exact differential equations,
Symbolic operations. Linear differential equations with constant coefficients. Differential
equations of the forms f (D) y = Sin bx, Cos ax, e
ax
V, where V is any function of x.
Homogeneous linear equations.
Unit IV:
Miscellaneous form of differential equations. First order higher degree equations solvable
for x,y,z,p. Equations from which one variable is explicitly absent, Clairut’s
form,  equations  reducible  to  Clairut’s   form. Legendre  polynomials.  Recurrencerelation and differential equation satisfied by it. Bessel functions, recurrence relation and
differential equation.
Books Recommended:
             1. S.D. Chopra and M.L. Kochar, - Integral Calculus.
             2. Shanti Narayan, -Integral Calculus.
             3. E.G. Philips, - Introduction to Differential Equations.
             4. Schuam Series, - Differential Equations.
             5. Shanti Narayan, - Vector Calculus.
Suggested Readings:
             1. T.M. Apostol, - Calculus.
             2. H.T.H. Piaggo – Differential Equations.


                                 MATRIX THEORY AND VECTOR CALCULUS


AMM-06                                                                                                     Marks: 50
Unit I:

Definition of a Determinant, Symmetric, Skew-symmetric, Hermition and skew-Hermition
matrices, Diagonal, scalar and triangular matrices, sum of matrices and properties of the
addition composition. Representation of a square matrix as a sum of a symmetric
(Hermition) and a skew-symmetric (Skew-Hermition) matrix. Representation of a square
matrix in the form of P + iQ, where P and Q are both Hermition.
Product pf matrices. Transpose of the product of two matrices and its generalization to
several matrices. Associative law for the product and Distributive law of matrices. Adjoint
of a square matrix A and relation A(adj.A)= (adj.A) =  AI, Inverse of a square matrix.
Reversal law for the inverse of a product of two matrices and its generalization to several
matrices.
Matrix polynomials, Characteristic and minimal equations of matrix. Cayley Hamilton
theorem. Rank of a matrix. Elementary row and rank of matrix by elementary
transformations.
Unit II:

Reduction of matrix to normal form. Elementary matrices. Every non-singular matrix is a
product of elementary matrices. Employment of only row (column) transformations. The
rank of a product of two matrices. Linear dependence and linear independence of column
(row) vectors. Linear combination of vectors.
The columns of a matrix A are linearly dependent iff there exists vector X≠ 0 such that Ax
= 0. The columns of a matrix A of order mn are linearly dependent iff rank of A is less
than n. The matrix A has rank r iff it has r linearly independent columns where as any s
columns, s>r are linearly dependent. Analogous results for rows. Linear homogeneous and
non-homogeneous equations. The equation AX= 0 has a non-zero solution iff rank of A is
less than n, the number of its columns. The number of linearly independent solutions of
the equation AX= 0 is (n-r) where r is the rank of mn matrix A. The equation AX=B is
consistent iff the two matrices A and A:B are of the same rank.Unit III:
Directed line segments, vectors and scalars, equality of two vectors, Algebra of vectors
addition, Multiplication with scalars, Laws of addition, Commutativity, associativity,
Magnitude of vectors, Difference of two vectors, a vector equation, Transformation of
coordinates of a vector, Linearly independent and Dependent System pf vectors, Scalar
product, Length of a vector and the angle between two vectors in terms of scalar products,
Properties of scalar products, Distributivity, Vector product, some properties of vector
products,
           The formula (a  b)  c = (a . c). (b . c) a.

Unit IV:
Scalar and vector product of three and four vectors, Reciprocal vectors, vector of functions
of a single scalar variable, limit of a vector function, Continuity, Vector differentiation,
Gradient, Divergence and Curl, Vector Integration, Guass’s Divergence Theorem, Green’s
Theorem and Stokes Theorem.
Books Recommended:

           1. A text book of Matrices, Shanti Narayan.
2. A text book of Matrices, Franz E. Hahan.
3. Vector Algebra, Shanti Narayan.
4. Vector Calculus, Shanti Narayan.
5. Vector Calculus, Schaum Series

•                                            B.A/B.Sc. (Three Years Degree Course)

                                                              In Applied Mathematics
                                                 B.A/B.SC. (3rd Year) W.e.f. 2006

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