| Paper number and name | Marks for internal evaluation | Marks for external evaluation | Total Marks | Work Load |
| Paper - III Advanced Calculus |
45 | 105 | 150 | 4 periods each of 55 minutes per week |
| Paper - IV A & B Theory and Practical |
Any one of the following | |||
| (1) Theory : Numerical Analysis & Theory of equations
(A)
Practical : (B) |
30
15 |
70
35 |
100
50 |
3 periods each of 55 minutes per week
A practical of periods each of 55 minutes per week. A patch contains maximum of 15 (fifteen students) |
| (2) Theory : Computer Oriented Numerical Methods (A)
Practical : (B) |
30
15 |
70
35 |
10
50 |
3 periods each of 55 minutes per week.
A practical of three periods each of 55 minutes per week. A practical batch contains maximum of 15 (fifteen students) |
| Paper - V Linear Algebra |
45 | 105 | 150 | 4 periods each of 55 minutes per week. |
Note : Only valued functions are to be considered.
Unit - 1 Functions of several variable :
Their limits and continuity, partial derivatives, differentiability and differentials, chain rule, differentials and
derivatives of higher orders, conditions for commutatively of independent variables in higher derivatives, derivatives of implicitly defined functions, differentiation along a curve, Euler's theorem on homogeneous functions.
Unit - 2 Applications of partial derivative :
Extreme of functions of several variables, Lagrange's method of undetermined multipliers, Tayor's and
Maclaurin's expansions for functions of several variables (proofs of two variables only), curvature in cartesian coordinates, singular points for plane curve (point of inflexion for plane algebraic curve) Indeterminate form, L-Hospital les.
Unit - 3 Integrals :
Upper and lower Riemann integral for a function of one variable defined over defined over a closed interval [a,b], Riemann integral and its properties leading to fundamental theorem of calculus, Mean value theorems, change of variable, repeated integration.
Unit - 4 Beta, Gamma functions :
Beta, Gamma functions, relations between them and simple properties and applications (sterling approximation is to be assumed), double integral, integral on nonrectangular regions, transformation to polar coordinates, triple integral, transformation to polar and cylindrical coordinates.
Unit - 5 Gradient of a scalar :
Divergence and curl of a vector, line integral surface integral, Stroke's, Green's and Gauss's theorems, and related problems.
Reference Books :
1. Advanced Calculus by D.V. Widder Prentice Hall, New Delhi.
2. Advanced Calculus Vol. I & II by T.M. Apostol Blaisdoll.
3. Advanced Calculus by R.C. Buck Macmillan
4. Advanced Calculus by S. Kaplan Addison - Wesley.
5. Calculus by Michel Spival
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Unit - 1 The real of an equation :
Finding an approximate value of a real root, graphical method, method of bisection, method of false position,
method of iteration, special iterative form the Newton - Raphson method, iterative form of square root, reciprocal and reciprocal squared root, roots of polynomial equations, Gauss Seidel method.
Synthetic division, derivative of a polynomial by synthetic division, Brige-Vieta method, Horner's method.
Unit - 2 Finite differences table and theory of Interpolation :
Descending, difference, ascending differences, symbolic operator, difference of polynomial, factorial polynomial, Gregory - Newton's forward and backward interpolation formula with remainder term.
Unit - 3 Central differences Interpolation Formula :
Gauss, forward and backward interpolation formula, Bessel's interpolation formula, Sterling interpolation formula, Laplace - Everett's interpolation formula, Lagrange's interpolation formula for equal and unequal intervals.
Divided differences. Newton's divide difference interpolation formula.
Unit - 4 Numerical differentiation and Integration :
Quadrature formula, symmetrical formula, simpom's rule, numerical solution of an ordinary differential equation, method of starting the solution, Taylor's method, Picard's method, Euler's method. Range - Kutta method.
Unit - 5 Theory of Equations :
Transformation of equations, relations between coefficients and roots, solutions of cubics and quadratics (Ferrari's and Cardan's methods), invariants G,H,I,J.
Reference Books :
1. Numerical Analysis by Kuns, McGraw-Hill
2. Methods in Numerical Analysis by K.W. Nelson, Macmillan.
3. Principles of Numerical Analysis by Householder, McGraw-Hill.
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Unit - 1 (a) Tracing of Graphs :
(i) conics (ii) trigonometric and inverse trigonometric functions (iii)
exponential, logarighmic and
hyperbolic functions and (iv) the algebraic curves cycloid, catenary and cardiod.
(b) On location of real root of an equation :
(i) method of false position (ii) method of iteration (iii) Horner's method (iv) Method of bisection.
Unit - 2 Interpolation :
(i) Gregory - Newton (Forward and Backward) formula. (ii) Central differences,
Gauss (Forward and
Backward) formulas. (iii) Bessel, Sterling, Everettee and Langrange's
interpolation. (iv) Newton's
divided difference formula.
Unit - 3 Differential equations, differentiation and integration :
(i) Differentiation (1st, 2nd order). (ii) Taylor's method
(iii) Picard's method (iv) Euler's method (v)
Runge-Katta's method (vi) Simpson's 1/3 and 3/8 rule.
Note : The use of the standard text book on Numerical Analysis may be permitted in of practical
examinations.
Tracing to Graphs :
1. (a) (i) Parabola (ii) Ellips (iii) Hyperbola (iv) Circle : Odd centre, radius (v) Pair of straight lines.
(b) Trigonometric and Inverse trigonometric functions : (i) sinx (ii) cosx (iii) tanx (iv) cotx (v) secx
(iv) cosecx (vii) sin-1 x (viii) cot-1 x (ix) tan-1 x (x) cot-1 x (xi) sec-1 x (xii) cosec-1 x
(c) Exponential, Logarithmic and Hyperbolic functions :
(i) ex (ii) ax (iii) logex (iv) coshx (v) tanhx (vi) cothx (ix) sechx (x) coceshx
(d) Algebraic curves :
(i) cycloid (ii) cardiod (iii) asroid (iv) catenery.
2. To find approximate value of a real root of an algebraic equation of f(x) = 0 by
(i) graphical method (ii) analytical method.
3.
To improve to approximation of a real root of an algebraic equation f(x) = 0 to
the required degree of
accuracy by the method of false
position.
4.
To obtain better approximation of a real root of an algebraic equation f(x) = 0
by the Newton-Raphson
method.
5.
To obtain better approximations of a real root of algebraic equation f(x) = 0
when expressed in the form
g(x) = h(x) from an integral or a root by
the method of iteration.
6.
To calculate (i) the square root (ii) the reciprocal and (iii) the reciprocal
square root of a given non-zero
positive number by Newton-Raphson formula.
7. (i) To compute the quotient Q and the remainder R when Pn (x) is divided by PI (x) and (ii) to determine
the derivatives of certain value of the argument by synthetic division. Also (iii) to find a real root of
p(x) = 0 by Birge - Vieta method.
8. (i) To write an equation with given roots each diminished by certain number and (ii) to calculate to
certain number of decimal places the roots of P(x) = 0 by Honer's method.
9. (i) To find and correct (by means of differences) the error in the given entries. (ii) To write a polymial in
terms of factorials and to determine the values of the factorials. (ii) to fit cubic polynomial passing
through four given points.
10. (i) To determine entry corresponding to certain argument from given tabular values x by Gregory-Newton
formula for forward interpolation. (ii) To calculate entry corresponding to
certain argument from given
tabular value x by Gregory-Newton formula
for backward interpolation. (iii) To compute entry corres-
ponding to certain argument from given
tabular value x by extrapolation.
11. (i) To determine entry corresponding to certain argument from given tabular values by forward
interpolation formula of Gauss. (ii) Calculate entry corresponding to certain
argument from given tabular
value by the backward interpolation formula
of Gauss.
12.
(i) To determine entry corresponding to certain argument from given tabular
values by Strilling's formula.
(ii) To calculate entry corresponding to
certain argument from given tabular values by the Bassel's
interpolation formula. (iii) To compute
entry corresponding to certain argument from given tabular values
by the Laplace-Everett's formula.
13.
(i) To determine entry corresponding to certain argument from given tabular
values by Largrange's
Interpolation (ii) To calculate entry
corresponding to certain argument from given tabular values by the
Newton's divided difference formula. (iii)
To compute entry corresponding to certain argument from given
tabular values either by Largrange's
general interpolation formula or by the method of successive
approximations.
14. (i) To determine first and second derivatives at certain argument from given tabular values by G.N.
Formula. (iii) To calculate first and second derivatives at certain argument
from given tabular values by
Stirling's Formula.
15.
To evaluate the definite integral by (i) trapezoidal rule (ii) Simpson's 1/3
rule (iii) Simpson's 3/8 rules (iv)
Weddle's rule.
16. To solve a differential with boundary conditions by (i) Taylor's Method (ii) Picard Method (iii) Euler's
Method (iv) Modified Method of Euler (v) Rung, Heun and Kutta Method.
Note : (i) A batch of 20 students shall be unit of practical.
(ii) The time - duration of a practical shall be the same as that of three periods each of 55 minutes
per week.
(iii) 25 practical are to be done in a year.
Unit - 1 Introduction to Programming in C :
Data types, Arithmetic and input/output instructions. Decisions control structures. Logical and conditional operators, Loop, case control structures, functions. Recursions, Preprocessors, Arrays, Puppetting of strings,
structures, pointers, file formatting.
Unit - 2 The real root of an equation :
Finding an approximate value of a real root, graphical method, method of bisection, method of false position, methods of iteration, special iterative form the Newton - Raphson method, iterative form of square root, reciprocal and reciprocal square roots of polynomial equations, Gauss Seidel method.
Synthetic division, derivative of a polynomial by synthetic division, Brige-Vieta method, Horner's method.
Unit - 3 Finite differences table and theory of Interpolation :
Descending difference, ascending differences, symbolic operator, difference of polynomial, factorial polynomial,
Gregory - Newton's forward and backward interpolation formula with remainder term.
Unit - 4 Central differences Interpolation Formula :
Gauss forward and backward interpolation formula, Bessel's interpolation formula, Sterling interpolation formula,
Laplace - Everett's interpolation formula, Lagrange's interpolation formula for equal and unequal intervals.
Divided differences. Newton's divide differences interpolation formula.
Unit - 5 Numerical differentiation and Integration :
Quadrature formula, symmetrical formula, simpom;s rule, numerical solution of an ordinary differential equation, method of starting the solution, Taylor's method, Picard's method, Euler's method. Range - Kutta method.
References Books :
1. Numerical Analysis by Kunz, McGraw-Hill Publication.
2. Numerical Analysis and Computational procedures by S.A. Mollah, New Central Book Agency, Culcutta.
3. Computer oriented Numerical Methods by R.S. Salaria, Khanna Book Publication, New Delhi.
4. Numerical Methods by E. Balaguruswamy.
5. Programming in Ansi-C, 2nd Edition by E. Balaguruswamy, Tata McGraw-Hill.
6. Elementary Numerical Analysis by S.S. Sastry, Prentice Hall of India, New Delhi.
7.
Numerical Methods for Mathematics, Science and Engineering by John H. Mathews,
Prentice Hall of India,
New Delhi.
1. Bisection Method : To find a real root of f(x) = 0
2. Fixed point iteration : To find a solution of x=g(x)
3. Method of false position : To find a root of f(x)=0
4. Newton Raphson Iteration :
Pi+1 = Pi - £(Pi) ; i=1,2,3....
-----
£(Pi)
5. Lagrange Interpolation.
6. Newton's divided interpolation.
7. Differentiation using limits.
8. Differentiation using N+1 nodes.
9. Differentiation using extrapolation.
10. Composite Trapezoidal Rule.
11. Composite Simpson Rule.
12. Euler's Method : To find approximation solution of dy/dx = f(x,y).
13. Taylor method of order 4.
14. Range-Kutta Method of order 4.
15. Golden search for a minimum.
16. Evaluation of a Taylor's series.
17. Polynomial Calculation - Synthetic division.
Note : 1. Each practical batch contains maximum of 10 students.
2. Practical Exam will be of 35 Marks.
3. One practical per week of 3 periods of 55 minutes.
4. Marks distributions :
(i) 5 Marks for Journal & Viva-voca
(ii) Students must have to write and run any two programms for given numerical
method - 30
marks. (15 Marks for each programme.)
Unit - 1
Linear space, subspaces, basic, dimension of a finite dimensional linear space and related standard results. Dimension theorem.
Linear transformations, range and kernel of a linear map, the spaces L(U,V) of linear mappings. Rank-Nullity
theorem, nonsingular linear transformations, Isomorphic vector spaces, operator equations.
Unit - 2
Matrix of a linear transformation, L.T. associated with a matrix, the dimension of L(U.V) and its determination.
Rank and nullity of a matrix, inevertibility of nxn matrices, types of matrices.
Elementary row operations and row reduction to row canonical form, application to the solution of a system of linear equations, discussion of the 3 cases when (I) there is no solutions, (II) There is a unique solution (III) there are infinite number of solutions, to find the inverse of a matrix by row reduction, similar matrieces.
Unit - 3
Dual space, dual basis, dual of a dual, annihilators, bilinear forms, matrix representations, symmetric and skew
symmetric bilinear forms.
Determinants and other invariants of a linear transformation, (Det AB = (Det A) (Det B), similar matrices have
the same determinant, determinant of a L.T. Trace of a matrix, trace of a L.T.).
Eigenvalues and eigenvectors of a linear transformation, diagonalization and eigenvectors, characteristics polynomial, Cayley-Hamilton theorem, minimal polynomial deducations.
Unit - 4
Inner product vector spaces, Cauchy - Schwarz inequality, angle between two vectors, orthogonality, Gram -
Schmidt's algorithm, orthonormal basis, adjoin of a linear transformation, special linear mapping and matrices
(Hermitian, unitary, orthogonal, normal)
Unit - 5
Special theorem for symmetric operators, symmetric real matrices, diagonolization of 2 x 2 and 3 x 3 real
matrices, (via orthogonal transformations, pivotal method.)
Geometrical Applications :
Quadratic forms associated with linear mappings, identifying quadratic curves in R2 quadratic surfaces in R3,
full linear group, group of transformations.
Note : All Units carry equal marks.
Reference Books :
1. An introduction to Linear Algebra by V. Krishnamurthy et al.
2. Topics in Algebra by I.N. Herstein, Vikas Publications, 2nd Edition.
3. Linear Algebra by S.K. Berberian, Oxford University Press.
4. Linear Algebra Problem Book by P.R. Holmos, Cambridge University Press.
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