The Present Book Differential Equations Provides A Detailed Account Of The Equations Of First Order And The First Degree, Singular Solutions And Orthogonal Trajectories, Linear Differential Equations With Constant Coefficients And Other Miscellaneous Differential Equations.It Is Primarily Designed For B.Sc And B.A. Courses, Elucidating All The Fundamental Concepts In A Manner That Leaves No Scope For Illusion Or Confusion. The Numerous High-Graded Solved Examples Provided In The Book Have Been Mainly Taken From The Authoritative Textbooks And Question Papers Of Various University And Competitive Examinations Which Will Facilitate Easy Understanding Of The Various Skills Necessary In Solving The Problems. In Addition, These Examples Will Acquaint The Readers With The Type Of Questions Usually Set At The Examinations. Furthermore, Practice Exercises Of Multiple Varieties Have Also Been Given, Believing That They Will Help In Quick Revision And In Gaining Confidence In The Understanding Of The Subject. Answers To These Questions Have Been Verified Thoroughly. It Is Hoped That A Thorough Study Of This Book Would Enable The Students Of Mathematics To Secure High Marks In The Examinations. Besides Students, The Teachers Of The Subject Would Also Find It Useful In Elucidating Concepts To The Students By Following A Number Of Possible Tracks Suggested In The Book.
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
Algebra Unit 1 0. Historical Background .... i-xvi 1. Linear Dependence and Independence of Row and Column Matrices and Rank of Matrix .... 1-58 2. Characteristic Equation of a Matrix, Eigen Values and Eigen Vectors .... 59-86 Unit 2 3. Cayley-Hamilton Theorem .... 87-97 4. Application of Matrices to a System of Linear Equation .... 98-125 Vector Analysis Unit 3 5. Product of Four Vectors and Reciprocal Vectors .... 126-155 6. Vector Differentiation .... 156-174 7. Gradient, Divergence and Curl .... 175-237 Unit 4 8. Vector Integration .... 238-250 9. Theorem of Gauss, Theorem of Green and Stoke’s Theorem (Without Proof); and Problems Based on them .... 251-300 10. Application to Geometry .... 301-356 Geometry Unit 5 11. General Equation of Second Degree and Tracing of Conics .... 357-407 12. System of Conics .... 408-432 13. Cone .... 433-485 14. Cylinder and its Properties .... 486-504
The Present Book Coordinate Geometry Of Two Dimensions Aims At Providing The Students With A Detailed Study Of Polar Coordinates, Polar Equations Of A Straight Line And A Circle, Polar Equations Of Conics, General Equation Of Second Degree And System Of Conics The Topics Included In The Ugc Syllabus.Primarily Meant For Students Of B.Sc./B.A. Of Several Indian Universities, The Book Exactly Covers The Prescribed Syllabus. It Neither Includes The Irrelevant Nor Escapes The Essential Topics. Its Approach Is Explanatory, Lucid And Comprehensive. The Analytic Explanation Of The Subject Matter Is Very Systematic Which Would Enable The Students To Assess And Thereby Solve The Related Problems Easily. Sufficient Number Of High-Graded Solved Examples Provided In The Book Facilitate Better Understanding Of The Various Skills Necessary In Solving The Problems. In Addition, Practice Exercises Of Multiple Varieties Will Undoubtedly Prove Helpful In Quick Revision Of The Subject. The Figures And Also The Answers Provided In The Book Are Accurate And Verified Thoroughly.A Proper Study Of The Book Will Definitely Bring To Students A Brilliant Success. Even Teachers Will Find It Useful In Elucidating The Subject To The Students Of Mathematics.
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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