Unit-I 0. Historical Background .... 1-4 1. Groups and Their Basic Properties .... 1-65 2. Subgroups .... 66-80 3. Cyclic Groups .... 81-93 4. Coset Decomposition, Lagrange’s and Fermat’s Theorem .... 94-113 5. Normal Subgroups .... 114-125 6. Quotient Groups .... 126-131 Unit-II 7. Homomorphism and Isomorphism of Groups, Fundamental Theorem of Homomorphism .... 132-151 8. Transformation and Permutation Group Sn (n < 5), Cayley’s Theorem .... 152-186 9. Group Automorphism, Inner Automorphism, Group of Automorphisms .... 187-206 Unit-III 10. Definition and Basic Properties of Rings, Subrings .... 207-232 11. Ring Homomorphism, Ideals, Quotient Ring .... 233-259 12. Polynomial Ringh .... 260-296 13. Integral Domain .... 297-310 14. Field .... 311-330 Unit-IV 15. Definition and Examples of Vector Space, Subspaces, Sum and Direct sum of Subspaces, Linear Span, Linear Dependence, Linear Independence and Their basic Properties .... 331-360 16. Basis, Finite Dimensional Vector Space and Dimension (Existence Theorem, Extension Theorem, Inoriance of the number of Elements), DImension of sum of Subspaces, Quonient Space and It Dimension .... 361-380 Unit-V 17. Linear Transformation and Its Representation as a Matrix .... 381-403 18. Algebra of Linear transformations, Rank-Nullity Theorem, Change of basis, Dual Space, Bi-dual Space and Natural Isomorphism Adjoint of a Linear Transformation .... 404-438 19. Eigen-Values and Eigen-Vectors of a Linear Transformation, Diagonalization .... 439-472
Unit-I 0. Historical Background .... i-iii 1. Field Structure and Ordered Structure of R, Intervals, Bounded and unbounded sets, Supremum and infimum, Completeness in R, Absolute value of a real Number .... 1-33 2. Sequence of Real Numbers, Limit of a Sequence, Bounded and Monotonic Sequences, Cauchy’s General Principle of Convergence, Algebra of Sequence and Some Important Theorems .... 34-80 Unit-II 3. Series of non-negative terms, Convergence of positive term series .... 81-146 4. Alternating Series and Leibrintr’s test, Absolute and conditional convergence of Series of real Terms .... 147-163 5. Uniform Continuity .... 164-185 6. Chain Rule of Differentiability .... 186-202 7. Mean Value Theorems and Their Geometrical Interpretations .... 203-228 Unit-III 8. Limit and continuity of functions of two variables .... 229-256 9. Change of Variables .... 257-280 10. Euler’s Theorem on Homogeneous Functions .... 281-294 11. Taylor’s Theorem For functions of two Variables .... 295-307 12. Jacobians .... 308-337 13. Maxima and Minima of Functions of Two Variables .... 338-354 14. Lagrange’s Multipliers Method .... 355-367 15. Beta and Gamma Functions .... 368-395 Unit-IV 16. Partial Differential Equations of The first order .... 396-415 17. Lagrange’s Solution .... 416-440 18. Some Special types of equations which can be solved easily by methods other than the general method .... 441-462 19. Charpit’s General Method .... 463-474 20. Partial Differential Equation of Second and Higher Order .... 475-485 Unit-V 21. Classification of Partial Differential Equations of Second Order .... 486-494 22. Homogeneous and Non-homogeneous Partial Differential Equations of Constant coefficients .... 495-541 23. Partial Differential Equations Reducible to Equtions with Constant Coefficients .... 542-551
Unit-I 0. Historical Background .... i-iii 1. Field Structure and Ordered Structure of R, Intervals, Bounded and unbounded sets, Supremum and infimum, Completeness in R, Absolute value of a real Number .... 1-33 2. Sequence of Real Numbers, Limit of a Sequence, Bounded and Monotonic Sequences, Cauchy’s General Principle of Convergence, Algebra of Sequence and Some Important Theorems .... 34-80 Unit-II 3. Series of non-negative terms, Convergence of positive term series .... 81-146 4. Alternating Series and Leibrintr’s test, Absolute and conditional convergence of Series of real Terms .... 147-163 5. Uniform Continuity .... 164-185 6. Chain Rule of Differentiability .... 186-202 7. Mean Value Theorems and Their Geometrical Interpretations .... 203-228 Unit-III 8. Limit and continuity of functions of two variables .... 229-256 9. Change of Variables .... 257-280 10. Euler’s Theorem on Homogeneous Functions .... 281-294 11. Taylor’s Theorem For functions of two Variables .... 295-307 12. Jacobians .... 308-337 13. Maxima and Minima of Functions of Two Variables .... 338-354 14. Lagrange’s Multipliers Method .... 355-367 15. Beta and Gamma Functions .... 368-395 Unit-IV 16. Partial Differential Equations of The first order .... 396-415 17. Lagrange’s Solution .... 416-440 18. Some Special types of equations which can be solved easily by methods other than the general method .... 441-462 19. Charpit’s General Method .... 463-474 20. Partial Differential Equation of Second and Higher Order .... 475-485 Unit-V 21. Classification of Partial Differential Equations of Second Order .... 486-494 22. Homogeneous and Non-homogeneous Partial Differential Equations of Constant coefficients .... 495-541 23. Partial Differential Equations Reducible to Equtions with Constant Coefficients .... 542-551
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