NORTH MAHARASHTRA UNIVERSITY JALGAON.

NORTH MAHARASHTRA UNIVERSITY JALGAON. Syllabus for S.Y.B.Sc. (Mathematics) With effect from June 013. (Semester system). The pattern of examination of theory papers is semester system. Each theory course is of 50 marks (40 marks external and 10 marks internal) and practical course is of 100 marks (80 marks external and 0 marks internal). The examination of theory courses will be conducted at the end of each semester and examination of practical course will be conducted at the end of the academic year. SEMESTER I STRUCTURE OF COURSES MTH-31 : Advanced Calculus MTH-3 (A) : Topics in Algebra OR MTH-3 (B) : Computational Algebra SEMESTER- II. MTH-41 : Complex Analysis MTH-4 (A) : Topics in Differential Equations OR MTH-4(B) MTH-03 : Differential Equations and Numerical methods : Practical Course based on MTH-31, MTH-3, MTH-41, MTH-4 1

SEMESTER I MTH -31: Advanced Calculus Unit- 1 : Functions of two and three variables Periods -15,Marks 10 1.1 Limits and Continuity 1. Partial Derivatives and Jacobians. 1.3 Higher order Partial Derivatives 1.4 Differentiability and Differentials 1.5 Necessary and sufficient conditions for Differentiability 1.6 Schwarz s Theorem 1.7 Young s Theorem. Unit- : Composite Functions and Mean Value Theorem Periods 15, Marks 10.1 Composite functions. Chain Rules.. Homogeneous functions..3 Euler s Theorem on Homogeneous Functions..4 Mean Value Theorem for Function of Two Variables. Unit -3 : Taylor s Theorem and Extreme Values Periods 15, Marks 10 3.1 Taylor s Theorem for a function of two variables. 3. Maclaurin s Theorem for a function of two variables. 3.3 Absolute and Relative Maxima and Minima. 3.4 Necessary Condition for extrema. 3.5 Critical Point, Saddle Point. 3.6 Sufficient Condition for extrema. 3.7 Lagrange s Method of undetermined multipliers. Unit -4 : Double and Triple Integrals Periods 15, Marks 10 4.1 Double integrals by using Cartesian and Polar coordinates. 4. Change of Order of Integration. 4.3 Area by Double Integral. 4.4 Evaluation of Triple Integral as Repeated Integral. 4.5 Volume by Triple Integral. Reference Books 1. Mathematical Analysis: S.C. Malik and Savita Arora. Wiley Eastern Ltd, New Delhi.. Calculus of Several Variables: Schaum Series. 3. Mathematical Analysis: T.M.Apostol. Narosa Publishing House, New Delhi, 1985 4. A Course of Mathematical Analysis: Shanti Narayan, S. Chand and Company, New Delhi.

MTH -3(A): Topics in Algebra Unit-1 : Groups 1.1 Definition of a group. 1. Simple properties of group. 1.3 Abelian group. 1.4 Finite and infinite groups. 1.5 Order of a group. 1.6 Order of an element and its properties. Unit- : Subgroups.1 Subgroups. Definition, criteria for a subset to be a subgroup.. Cyclic groups..3 Coset decomposition..4 Lagrange s theorem for finite group..5 Euler s theorem and Fermat s theorem. Unit -3 : Homomorphism and Isomorphism of Groups 3.1 Definition of Group Homomorphism and its Properties. 3. Kernel of Homomorphism and Properties. 3.3 Definition of Isomorphism and Automorphism of Groups. 3.4 Properties of Isomorphism of Groups. Unit-4 : Rings 4.1 Definition and Simple Properties of a Ring. 4. Commutative Ring, Ring with unity, Boolean Ring. 4.3 Ring with zero divisors and without zero Divisors. 4.4 Integral Domain, Division Ring and Field. Simple Properties. Reference Books 1. Topics in Algebra : I. N. Herstein.. A first Course in Abstract Algebra : J. B. Fraleigh. 3. University Algebra : N. S. Gopalkrishanan. Periods-15, Marks-10. Periods-15, Marks-10. Periods-15, Marks-10 Periods-15, Marks-10 3

MTH -3(B): Computational Algebra Unit-1 : Groups Periods-15, Marks-10. 1.1 Definition of a group. 1. Simple properties of group. 1.3 Abelian group. 1.4 Finite and infinite groups. 1.5 Order of a group. 1.6 Order of an element and its properties. Unit- : Subgroups Periods-15, Marks-10..1 Subgroups. Definition, criteria for a subset to be a subgroup.. Cyclic groups..3 Coset decomposition..4 Lagrange s theorem for finite group..5 Euler s theorem and Fermat s theorem. Unit -3 : Homomorphism and Isomorphism of Groups Periods-15, Marks-10 3.1 Definition of group homomorphism and its properties. 3. Kernel of homomorphism and properties. 3.3 Definition of isomorphism and automorphism of groups. 3.4 Properties of isomorphism of groups. Unit-4 : Group Codes Periods -15, Marks 10 4.1 Message, Word, Encoding Function, Code Word. 4. Detection of K or Fewer Error, Weight Parity, Check Code 4.3 The Hamming Distance, Properties of the Distance Function, The Minimum Distance of an Encoding function. 4.4 Group Codes. 4.5 (n, m) Decoding function, Maximum Likelihood Decoding Function. 4.6 Decoding Procedure for a Group Code Given by a Parity Matrix. Reference Books 1. Discrete Mathematical Structure : Bernard Kolman Robert C. Busby, Ross Prentice Hall and India New Delhi. Eastern Economy Edition.. Topics in Algebra : I.N. Herstein. 3. A first Course in Abstract Algebra : J. B. Fraleigh. 4. University Algebra : N. S. Gopalkrishanan. 4

SEMESTER II MTH -41 : Complex Analysis Unit-1 : Complex numbers 1.1 Algebra of complex numbers 1. Representation of complex numbers as an order pair 1.3 Modulus and amplitude of a complex number 1.4 Triangle inequality and Argand s diagram 1.5 De Moivre s theorem for rational indices 1.6 n th roots of a complex number Unit-: Functions of complex variables.1 Limits, Continuity, Derivative.. Analytic functions, Necessary and sufficient condition for analytic function..3 Cauchy Riemann equations..4 Laplace equations and Harmonic functions.5 Construction of analytic functions Unit-3 : Complex integrations 3.1 Line integral and theorems on it. 3. Statement and verification of Cauchy-Gaursat s Theorem. 3.3 Cauchy s integral formulae for f(a) and f (a) 3.4 Taylor s and Laurent s series. Unit-4 : Calculus of Residues 4.1 Zeros and poles of a function. 4. Residue of a function 4.3 Cauchy s residue theorem 4.4 Evaluation of integrals by using Cauchy s residue theorem 4.5 Contour integrations of the type Reference books 1. Complex Variables and Applications ; R.V.Churchill and J.W. Brown. Theory of Functions of Complex Variables : Shanti Narayan S. Chand and Company, New Delhi. 3. Complex variables : Schaum Series. 5

MTH-4(A): Topics in Differential Equations Unit-1 : Theory of ordinary differential equations 1.1 Lipschitz condition 1. Existence and uniqueness theorem 1.3 Linearly dependent and independent solutions 1.4 Wronskian definition 1.5 Linear combination of solutions 1.6 Theorems on i) Linear combination of solutions ii) Linearly independent solutions iii) Wronskian is zero iv) Wronskian is non zero 1.7 Method of variation of parameters for second order L.D.E. Unit- : Simultaneous Differential Equations.1 Simultaneous linear differential equations of first order. Simultaneous D.E. of the form.3 Rule I: Method of combinations.4 Rule II: Method of multipliers.5 Rule III: Properties of ratios.6 Rule IV: Miscellaneous Unit-3 : Total Differential or Pfaffian Differential Equations 3.1 Pfaffian differential equations 3. Necessary and sufficient condition for integrability 3.3 Conditions for exactness 3.4 Method of solution by inspection 3.5 Solution of homogenous equation 3.6 Use of auxiliary equations Unit-4 : Beta and Gamma Functions 4.1 Introduction 4. Euler s Integrals: Beta and Gamma functions 4.3 Properties of Gamma function 4.4 Transformation of Gamma function 4.5 Properties of Beta function 4.6 Transformation of Beta function 4.7 Duplication formula Reference Book- 1. Ordinary and Partial Differential Equation by M. D. Rai Singhania, S. Chand & Co.(11 th Revised Edition) 6

MTH-4(B): Differential Equations and Numerical methods Unit-1 : Theory of ordinary differential equations 1.1 Lipschitz condition 1. Existence and uniqueness theorem 1.3 Linearly dependent and independent solutions 1.4 Wronskian definition 1.5 Linear combination of solutions 1.6 Theorems on i) Linear combination of solutions ii) Linearly independent solutions iii) Wronskian is zero iv) Wronskian is non zero 1.7 Method of variation of parameters for second order L.D.E. Unit- : Simultaneous Differential Equations.1 Simultaneous linear differential equations of first order. Simultaneous D.E. of the form.3 Rule I: Method of combinations.4 Rule II: Method of multipliers.5 Rule III: Properties of ratios.6 Rule IV: Miscellaneous Unit-3 : Total Differential or Pfaffian Differential Equations 3.1 Pfaffian differential equations 3. Necessary and sufficient condition for integrability 3.3 Conditions for exactness 3.4 Method of solution by inspection 3.5 Solution of homogenous equation 3.6 Use of auxiliary equations Unit-4 : Numerical Methods for solving ODE 4.1 Picard s method 4. Taylor series method 4.3 Modified Euler s method 4.4 Runge-Kutta method of fourth order 4.5 Milne s Method 4.6 Adam Bashforth method Reference Book- 1. Ordinary and Partial Differential Equation by M.D.Rai Singhania, S.Chand & Co.(11 th Revised Edition). Numerical Analysis By S.S. Sastry 7

NORTH MAHARASHTRA UNIVERSITY JALGAON S.Y.B.Sc. Mathematics Practical Course MTH-03 Based on MTH-31, MTH-3, MTH-41, MTH-4. (With effect from June 013) Index Practical Title of Practical No. 1 Functions of two and three variables Composite Functions and Mean Value Theorem. 3 Taylor s Theorem and Extreme Values. 4 Double and Triple Integrals 5 Groups 6 Subgroups 7 Homomorphism and Isomorphism of Groups 8(A) Rings 8(B) Group Codes 9 Complex Numbers 10 Functions of complex variables 11 Complex Integration 1 Calculus of Residues 13 Theory of Ordinary Differential Equations 14 Simultaneous Differential Equations 15 Total Differential or Pfaffian Differential Equations 16(A) Beta and Gamma functions 16(B) Numerical Methods for solving Ordinary Differential Equations 8

Practical No. 1 : Functions of two and three variables 1. a) Evaluate along the path y = x b) Prove that does not exists.. a) b) Show that the function 3. a) If b) 4. Let 5. a) b) 9

Practical No. : 1. a) Composite Functions and Mean Value Theorem.. a) If b) Show that. b) If 3. a) If. b) If. 4 a) If b) 5 If then show that used in mean value theorem applied to the points (, 1) and (4, 1) satisfies the quadratic equation, Practical No. 3 : Taylor s Theorem and Extreme Values. 1. a) Expand in the powers of using Taylor s expansion. b) Expand about (1, 1) up to third degree terms using Taylor s expansion. a) Show that b) Expand in the powers of x and y up to third degree terms 3. Expand upto the third degree terms and hence find 4 a) Find the maxima and minima of b) Investigate the maxima and minima of 5 a) Classify the critical points of b) Find the shortest distance of from the origin.. 10

1. Evaluate. Evaluate a a y Practical No. 4 : Double and Triple Integrals a x y dydx 0 0 ( x y ) dxdy, over the region in the positive quadrant for which 3. Change the order of the integration 4. a) Evaluate a 3a x 0 x 4a f ( x, y) dxdy x x y xe dxdy b) Find the area of the ellipse 0 0 5. a) Evaluate ( x y z) dxdydz, where R : R b) Find the volume of the sphere by using triple integral. Practical No. 5 : Groups 1. a) Show that the set of all positive rational numbers forms an abelian group under the binary operation defined by b) Show that G = -{-1} is an abelian group under the binary operation, a, b G.. a) Let G = { (a, b) : a, b, a 0 }. Show that (G, ) is a non-abelian group, where (a, b) (c, d) = (ac, bc+d). b) Prove that is a group under matrix multiplication. 3. a) In (, ), find i) ii) iii) iv) 0( v) 0( b) In (, ), find i) ii) iii) iv) 0( v) 0( 4. If G is a group such that a = e, for all a, b G, then show that G is abelian. 5. If in a group G, a 5 = e and aba 1 = b for a, b G, then find order of b. Practical No. 6 : Subgroups 1. If H is a subgroup of a group G and x G, then show that xhx -1 = { xhx -1 : h H } is a subgroup of G.. a) Let G be a group of all nonzero complex number under multiplication. Show that H = { a + ib G : a + b = 1 } is subgroup of G. b) Determine whether and are subgroups of 1. 3. a) Show that is a cyclic group. Find all its generators, all its proper subgroups and the order of every element. b) Find all subgroups of. 4. a) Let. Find all subgroups of G. b) Let G = 6 and be subgroup of G. Find all right and left cosets of H. 5. a) Show that every proper subgroup of a group of order 77 is cyclic. b) Find the remainder obtained when 33 19 is divisible by 7. 11

Practical No. 7 : Homomorphism and Isomorphism of Groups 1. Prove that the mapping f : 0 such that f(z) = e z is a homomorphism of the additive group of complex numbers onto the multiplicative group of nonzero complex number. What is the kernel of f.. Let G = {(a, b) : a, b } be a group under addition defined by (a, b) + (c, d) = (a + c, b + d). Show that the mapping f : G defined by f(a, b) = a, (a, b) G, is a homomorphism and find its kernel. 3. a) Let G be a group of all matrices of the type under matrix multiplication and G be a group of nonzero complex numbers under multiplication. Show that f : G G defined by is an isomorphism. b) If G = { 1, -1, i, -i } is a group under multiplication and is a group under multiplication modulo 10, then show that G and are isomorphic. 4. a) Let G be a group and G. Show that f : G G defined by f(a) = xax 1, a G is an automorphism. b) Let * be the multiplicative group of nonzero complex numbers and * be the multiplicative group of nonzero real numbers. Define : * * by for z *. Show that is a homomorphism. Is an isomorphism? Explain. 5. Let G be a group and f : G G be a map defined by f(x) = x -1, G. Prove that a) If G is abelian, then f is an isomorphism. b) If f is a group homomorphism, then G is abelian. Practical No. 8(A) : Rings 1. a) Let be the set of even integers and is a binary operation on is defined as, where ab is ordinary multiplication. Show that (, +, ) is a commutative ring, where + is ordinary addition in. b) For a, b define a b = a + b 1 and a b = a + b ab. Show that (,, ) is commutative ring with identity element.. Show that = {a + ib : a, b }, the set of Gaussian integers, forms an integral domain under usual addition and multiplication of complex numbers. 3. Show that is an integral domain. 4. If R is a ring in which x = x, for every x R, then prove that i) x + x = 0 ii) x + y = 0 x = y iii) R is commutative. 5. a) In the ring, find i) the zero element of the ring ii) units in iii) additive inverse of 3 iv) multiplicative inverse of 3 v) -8 3 vi) 4 (-8) b) Which of the following are fields : i) ii) iii) iv) v). Justify your answer. 1

Practical No. 8(B) : Group Codes 1. a) Consider the encoding function defined by,,,,,,, i. Find the minimum distance of e. ii. How many errors will e detect? b) Show that the encoding function defined by,,,,,,, is a group code. Also find the minimum distance of e.. Compute: i. ii. 3. Let be a parity check matrix. Determine the group code. 4. Consider the parity check matrix : Determine the coset leaders for. Also compute the Syndrome for each coset leader and decode the code relative to maximum likelihood decoding function. 5. Let the decoding function defined by, where has atleast two 1 s. has less than two 1 s,, then determine, where i. ii. 13

Practical No.9 : Complex Numbers 1. Find the modulus argument form (polar form) of following complex numbers : a) 3 + 4i b) - i. If represent the vertices of an equilateral triangle, then prove that 3. a) If, then prove that b) Prove that = 1, if either 4. If cos + cos + cos = 0 and sin + sin + sin = 0, then show that a) cos3 + cos3 + cos3 = 3 cos( + + ) and sin3 + sin3 + sin3 = 3 sin( + + ) b) cos + cos + cos = 0 = sin + sin + sin 5. a) Find all values of b) Solve the equation x 4 x 3 + x x + 1 = 0. Practical No. 10 : Functions of complex variables 1. a) Evaluate. b) Show that does not exist.. Discuss the continuity of, =, if at 3. Construct an analytic function and express it in terms of. 4. Prove that is harmonic and find its harmonic conjugate. Also find the corresponding analytic function. 5. is analytic, then prove that... 14

1. Expand in powers of for Practical No. 11 : Complex Integration b) c). Expand in Laurent s series valid for a) b) 3. Evaluate where C is a) The Straight line joining b) The straight line joining first and then 4. Verify Cauchy s theorem for taken round the unit circle 5. Use Cauchy s integral formula to evaluate: a) b).. Practical No. 1 : Calculus of Residues 1. Compute the residue at double poles of.. Evaluate by Cauchy s residue theorem, where C is a) The Circle b) The Circle 3. Evaluate by Contour integration :. 4. Apply Calculus of residues to evaluate. 5. Use Calculus of residues to evaluate.. 15

Practical No. 13 : Theory of Ordinary Differential Equations where 1. Show that y = 3e x + e -x - 3x is the unique solution of the initial value problem y 4y 1 x, y(0) 4, y (0) 1.. Find the unique solution of y 1where y(0) 1 and y (0). 3. Show that f(x,y) = xy satisfies the Lipschitz condition on the rectangle R: x 1, y 1 but does not satisfy the Lipschitz condition on the strip S : x 1, y. 4. Prove that sinx and cosx are solutions of y 4y 0and these solutions are linearly independent. 5. a) Show that e x sinx and e x cosx are linearly independent solutions of the differential equation y y y 0. What is the general solution? Find the solution y(x) if y(0) = and y (0) 0. b) Prove that 1, x, x are linearly independent. Hence, form the differential equation whose solutions are 1, x and x. Practical No. 14 : Simultaneous Differential Equations dx dy 1. a) Solve y 1, x 1, by the method of differentiation. dt dt dx t dy t b) Solve 5 x y e, x 3 y e, by operator method. dt dt dx dy dz dx dy dz. a) Solve b) Solve y x x y z y x xy xz 3. a) Solve adx bdy cdz ( b c ) yz b) Solve dx dy dz ( c a ) xz ( a b ) yx ( y z ) x ( z x ) y ( x y ) z 4. Solve dx dy dz y ( x y) z ( y z) x ( z x) 5. a) Solve dx dy dz dx dy dz b) Solve x y z xy xz y z z x x y 16

Practical No. 15 : Total Differential or Pfaffian Differential Equations 1. Show that the following differential equation is integrable and hence solve it: a) b) ( y z x ) dx xydy xzdz 0 yz x yz dx zx y zx dy xy z xy dz ( ) ( ) ( ) 0. Solve the following by the method of homogenous differential equations : a) ( y z) dx ( x y) dy ( y x z) dz 0 b) ( y yz) dx ( z xz) dy ( y xy) dz 0 3. Solve the following by the method of homogenous differential equations : a) b) ( xz yz) dx ( yz xz) dy ( x xy y ) dz 0 z dx z yz dy y yz xz dz ( ) ( ) 0 4. Solve the following by the method of auxiliary equations : a) b) c) 3 xz dx zdy yzdz 0 ( xz yz) dx ( yz zx) dy ( x xy y ) dz 0 ( y yz z ) dx ( z xz x ) dy ( x yx y ) dz 0 1. a) Evaluate Practical No. 16(A) : Beta and Gamma functions 6 x x e dx b) Prove that 0 1 9 3 945 n ax n 1. a) Prove that n 1.3.5...(n 1) b) Show that e x dx ( n>0 ) 0 n a a x a 1 3. a) Show that 0 x dx a 1 ( a 0) ax x a b) Show that e dx e a (log a) 1 3 16 3 4. a) Show that x(8 x ) dx b) Evaluate 0 3 5. Evaluate 1 0 0 x dx 1 x dx 0 6 1 x x n 17

Practical No. 16(B) : Numerical Methods for solving Ordinary Differential Equations 1. a) Using Picard s method of successive approximations obtain a solution upto third approximation of the differential equation dy dx x y with y(0) = 0. b) Using Picard s method of successive approximations obtain a solution upto third approximation of the differential equation dy dx 4 x y x with y(0) = 3.. a) From the Taylor s series for y(x), find y(0.1) correct upto four decimal places if y(x) satisfies dy dx x y and y(0) = 1. b) From the Taylor s series for y(x), find y(0.1) correct upto four decimal places if y(x) satisfies dy y 3e x and y(0) = 1 and compare with the exact equation. dx dy log ( x y), with the initial condition that y = 1 when x = 0, find y for x = 0., dx 3. a) Given 10 using modified Euler s method. dy with the initial condition that y = 1 when x = 0, find y for x = 0., dx b) Given x y, using modified Euler s method. 4. a) Using Runge-Kutta method of fourth order solve b) Find y() if y(x) is the solution of dy y x dx and y(1.5)=4.968, using Milne s method. 5. a) Find y(0.4) if y(x) is the solution of dy dx dy y x dx y x y(0.) = 1.3 and y(0.3) = 1.354, using Milne s method. b) Find y(1.4) if y(x) is the solution of with y(0) = 1 at x = 0.., assuming y(0) =, y(0.5) =.636, y(1) = 3.595 1 xy, assuming y(0) = 1, y(0.1) = 1.105, dy x (1 y), assuming y(1) = 1, y(1.1) = 1.33, dx y(1.) = 1.548 and y(1.3) = 1.979, using Adam-Bashforth method. 18