Engineering Mathematics
Index of the Lesson Plan |
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UNIT – I NUMBER THEORY – 12 HRS |
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| L-1 | Introduction, Conjectures, Theorems and Proofs |
| L-2 | Well ordering Principle, Principle of mathematical induction – strong and weak – problems |
| L-3 | Primes and Composite numbers, Fibonacci numbers, Lucas numbers - problems |
| L-4 | Sigma notation, Product notation, Binomial co-efficients – properties, problems |
| L-5 | Greatest integer function – properties, problems |
| L-6 | Computers exercises and review exercises |
| L-7 | Divisibility, proper divisor, non-trivial divisor, properties of divisibility, division algorithm |
| L-8 | Common divisors, Greatest common divisor, properties of Greatest common divisor, Greatest common divisor via Euclid’s algorithm, Linear form of GCD, problems |
| L-9 | Relatively prime numbers, Euclid’s lemma, pair wise relatively primes - problems |
| L-10 | Common multiple, LCM, properties of LCM |
| L-11 | Decimal representation of integers, binary representation of integers - problems |
| L-12 | Computer exercises and review exercises |
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UNIT – II PRIMES & CONGRUENCES – 12 HRS |
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| L-13 | Theorems on Composite and prime numbers, Twin primes |
| L-14 | Prime counting function, Asymptotic functions, prime number theorem |
| L-15 | Canonical factorization of a natural number, fundamental theorem of arithmetic |
| L-16 | Dirichlet’s theorem, Bertrand’s postulates, Goldbach’s conjectures, problems |
| L-17 | Test of Primality by trial division |
| L-18 | Computer exercises and review exercises |
| L-19 | Congruences and equivalence relations, equivalence class - problems |
| L-20 | Least positive residue, linear congruences - problems |
| L-21 | Chinese remainder theorem and problems |
| L-22 | Modular arithmetic: Fermat’s little theorem |
| L-23 | Fermat’s last theorem (Statement only), Wilson’s theorem and Fermat numbers |
| L-24 | Computer exercises and review exercises |
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UNIT – III ARITHMETIC FUNCTIONS, PRIMITIVE ROOTS |
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| L-25 | Arithmetic functions: sigma function, Tau function, Summation function, Sum of divisiors function, numbers of divisors function, problems |
| L-26 | Multiplicative function, totally multiplicative function, Dirichlet product – properties and theorems, perfect numbers and Mersenne primes |
| L-27 | Dirichelt inverse, Moebius function, Euler’s function, Euler’s theorem |
| L-28 | Primitive roots – Definition and properties, Primitive root (mod m) |
| L-29 | Index, Properties of indices – problems |
| L-30 | Computer exercises and review exercises |
| L-31 | Quadratic residue and non-residue (mod p) |
| L-32 | Euler’s criterion, properties of the Legendre symbol - problems |
| L-33 | Gauss Lemma and related theorems |
| L-34 | Law of quadratic reciprocity |
| L-35 | Solution of quadratic congruences, Algorithm for solving quadratic congruences (mod p) - problems |
| L-36 | Computer exercises and review exercises |
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UNIT -IV THE VECTOR SPACE RN
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| L-37 | Introduction to vectors, zero vector, Negative vector, Subtraction, Column Vectors |
| L-38 | Dot Product, Norm, Angle and Distance, Norm of a Vector in Rn |
| L-39 | Angle between Vectors, Distance between Points,
Introduction to Linear Transformations |
| L-40 | Composition of Matrix Transformations. |
| L-41 | Linear Transformations, Kernel and Range Terminology |
| L-42 | Transformations and systems of Linear Equations,
Homogeneous Equations, Non homogeneous Equations |
| L-43 | Many Systems, Coordinate Vectors, Notation, Change of Basis |
| L-44 | Isomorphisms, Matrix Representations of Linear Transformations |
| L-45 | Importance of Matrix Representations, Relations
between Matrix Representations |
| L-46 | Diagonal Matrix Representation of a Linear Operator |
| L-47 | Diagonalization of a matrix-matrix partitioning |
| L-48 | Problems on L-48 continued |
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UNIT -V NUMERICAL TECHNIQUES & |
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| L-49 | Gaussian Elimination, Comparison of Gauss-Jordan
and Gaussian Elimination |
| L-50 | Method of LU Decomposition, Construction of an LU decomposition of a Matrix |
| L-51 | Practical Difficulties in solving Systems of
Equations, The Condition Number of a Matrix |
| L-52 | Pivoting and Scaling Techniques |
| L-53 | Iterative methods for solving systems of Linear
Equations, Jacobi Method |
| L-54 | Gauss-Siedel Method |
| L-55 | Eigen values by Iteration, Connectivity of Networks |
| L-56 | Deflation Accessibility Index of a Network |
| L-57 | A Geometrical Introduction to Linear Programming,
A Linear Programming Problem |
| L-58 | Minimum Value of a Function, Discussion of the Method |
| L-59 | The simplex Method, Geometrical Explanation of
the Simplex Method |
| L-60 | Problems on L-59 continued |
SYLLABUS FOR TESTS
| Test – I Unit I: Number Theory Unit II: Primes & Congruences |
Lesson Numbers L-1 to L-20 |
| Test – II Unit II: Primes & Congruences Unit III: Arithmetic Functions, Primitive roots & Indices & Quadratic Congruences |
L-21 to L-36 |
| Test – III Unit IV: The vector space and Linear Transformation Unit V: Numerical Techniques & Linear Programming |
L-37 to L-60 |