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Engineering Mathematics, Volume I, Second Edition

Book Description

Engineering Mathematics Volume-I is meant for undergraduate engineering students. Considering the vast coverage of the subject, usually this paper is taught in three to four semesters. The two volumes in Engineering Mathematics by Babu Ram offer a complete solution to these papers.

Table of Contents

  1. Cover
  2. Title Page
  3. Contents
  4. Dedication
  5. Preface
  6. Symbols and Basic Formulae
  7. 1 - Sequences and Series
    1. 1.1 - SEQUENCES
    2. 1.2 - CONVERGENCE OF SEQUENCES
    3. 1.3 - THE UPPER AND LOWER LIMITS OF A SEQUENCE
    4. 1.4 - CAUCHY'S PRINCIPLE OF CONVERGENCE
    5. 1.5 - MONOTONIC SEQUENCE
    6. 1.6 - THEOREMS ON LIMITS
    7. 1.7 - SUBSEQUENCES
    8. 1.8 - SERIES
    9. 1.9 - COMPARISON TESTS
    10. 1.10 - D'ALEMBERI'S RATIO TEST
    11. 1.11 - CAUCHY'S ROOT TEST
    12. 1.12 - RAABE'S TEST
    13. 1.13 - LOGARITHMIC TEST
    14. 1.14 - DE MORGAN–BERIRAND TEST
    15. 1.15 - GAUSS'S TEST
    16. 1.16 - CAUCHY'S INTEGRAL TEST
    17. 1.17 - CAUCHY'S CONDENSATION TEST
    18. 1.18 - KUMMER'S TEST
    19. 1.19 - ALTERNATING SERIES
    20. 1.20 - ABSOLUTE CONVERGENCE OF A SERIES
    21. 1.21 - CONVERGENCE OF THE SERIES OF THE TYPE
    22. 1.22 - DERANGEMENT OF SERIES
    23. 1.23 - NATURE OF NON-ABSOLUTELY CONVERGENT SERIES
    24. 1.24 - EFFECT OF DERANGEMENT OF NON-ABSOLUTELY CONVERGENT SERIES
    25. 1.25 - UNIFORM CONVERGENCE
    26. 1.26 - UNIFORM CONVERGENCE OF A SERIES OF FUNCTIONS
    27. 1.27 - PROPERTIES OF UNIFORMLY CONVERGENT SERIES
    28. 1.28 - POWER SERIES
    29. EXERCISES
  8. 2 - Successive Differentiation, Mean Value Theorems and Expansion of Functions
    1. 2.1 - SUCCESSIVE DIFFERENTIATION
    2. 2.2 - LEIBNITZ'S THEOREM AND ITS APPLICATIONS
    3. 2.3 - GENERAL THEOREMS
    4. 2.4 - TAYLOR'S INFINITE SERIES AND POWER SERIES EXPANSION
    5. 2.5 - MACLAURIN'S INFINITE SERIES
    6. 2.6 - EXPANSION OF FUNCTIONS
    7. 2.7 - INDETERMINATE FORMS
    8. EXERCISES
  9. 3 - Curvature
    1. 3.1 - RADIUS OF CURVATURE OF INTRINSIC CURVES
    2. 3.2 - RADIUS OF CURVATURE FOR CARTESIAN CURVES
    3. 3.3 - RADIUS OF CURVATURE FOR PARAMETRIC CURVES
    4. 3.4 - RADIUS OF CURVATURE FOR PEDAL CURVES
    5. 3.5 - RADIUS OF CURVATURE FOR POLAR CURVES
    6. 3.6 - RADIUS OF CURVATURE AT THE ORIGIN
    7. 3.7 - CENTER OF CURVATURE
    8. 3.8 - EVOLUTES AND INVOLUTES
    9. 3.9 - EQUATION OF THE CIRCLE OF CURVATURE
    10. 3.10 - CHORDS OF CURVATURE PARALLEL TO THE COORDINATE AXES
    11. 3.11 - CHORD OF CURVATURE IN POLAR COORDINATES
    12. 3.12 - MISCELLANEOUS EXAMPLES
    13. EXERCISES
  10. 4 - Asymptotes and Curve Tracing
    1. 4.1 - DETERMINATION OF ASYMPTOTES WHEN THE EQUATION OF THE CURVE IN CARTESIAN FORM IS GIVENS
    2. 4.2 - THE ASYMPTOTES OF THE GENERAL RATIONAL ALGEBRAIC CURVE
    3. 4.3 - ASYMPTOTES PARALLEL TO COORDINATE AXES
    4. 4.4 - WORKING RULE FOR FINDING ASYMPTOTES OF RATIONAL ALGEBRAIC CURVE
    5. 4.5 - INTERSECTION OF A CURVE AND ITS ASYMPTOTES
    6. 4.6 - ASYMPTOTES BY EXPANSION
    7. 4.7 - ASYMPTOTES OF THE POLAR CURVES
    8. 4.8 - CIRCULAR ASYMPTOTES
    9. 4.9 - CONCAVITY, CONVEXITY AND SINGULAR POINTS
    10. 4.10 - CURVE TRACING (CARTESIAN EQUATIONS)
    11. 4.11 - CURVE TRACING (POLAR EQUATIONS)
    12. 4.12 - CURVE TRACING (PARAMETRIC EQUATIONS)
    13. EXERCISES
  11. 5 - Functions of Several Variables
    1. 5.1 - CONTINUITY OF A FUNCTION OF TWO VARIABLES
    2. 5.2 - DIFFERENTIABILITY OF A FUNCTION OF TWO VARIABLES
    3. 5.3 - THE DIFFERENTIAL COEFFICIENTS
    4. 5.4 - DISTINCTION BETWEEN DERIVATIVES AND DIFFERENTIAL COEFFICIENTS
    5. 5.5 - HIGHER-ORDER PARTIAL DERIVATIVES
    6. 5.6 - ENVELOPES AND EVOLUTES
    7. 5.7 - HOMOGENEOUS FUNCTIONS AND EULER'S THEOREM
    8. 5.8 - DIFFERENTIATION OF COMPOSITE FUNCTIONS
    9. 5.9 - TRANSFORMATION FROM CARTESIAN TO POLAR COORDINATES AND VICE VERSA
    10. 5.10 - TAYLOR'S THEOREM FOR FUNCTIONS OF SEVERAL VARIABLES
    11. 5.11 - APPROXIMATION OF ERRORS
    12. 5.12 - GENERAL FORMULA FOR ERRORS
    13. 5.13 - TANGENT PLANE AND NORMAL TO A SURFACE
    14. 5.14 - JACOBIANS
    15. 5.15 - PROPERTIES OF JACOBIAN
    16. 5.16 - NECESSARY AND SUFFICIENT CONDITIONS FOR JACOBIAN TO VANISH
    17. 5.17 - DIFFERENTIATION UNDER THE INTEGRAL SIGN
    18. 5.18 - MISCELLANEOUS EXAMPLES
    19. 5.19 - EXTREME VALUES
    20. 5.20 - LAGRANGE'S METHOD OF UNDETERMINED MULTIPLIERS
    21. EXERCISES
  12. 6 - Tangents and Normals
    1. 6.1 - INTRODUCTION
    2. 6.2 - EQUATION OF THE TANGENT AT A POINT OF A CURVE
    3. 6.3 - EQUATION OF THE NORMAL AT A POINT P(X1, Y1) OF A CURVE
    4. 6.4 - LENGTHS OF TANGENT, NORMAL, SUB-TANGENT AND SUBNORMAL AT ANY POINT OF A CURVE
    5. EXERCISES
  13. 7 - Beta and Gamma Functions
    1. 7.1 - BETA FUNCTION
    2. 7.2 - PROPERTIES OF BETA FUNCTION
    3. 7.3 - GAMMA FUNCTION
    4. 7.4 - PROPERTIES OF GAMMA FUNCTION
    5. 7.5 - RELATION BETWEEN BETA AND GAMMA FUNCTIONS
    6. 7.6 - DIRICHLET'S AND LIOUVILLE'S THEOREMS
    7. 7.7 - MISCELLANEOUS EXAMPLES
    8. EXERCISES
  14. 8 - Reduction Formulas
    1. 8.1 - REDUCTION FORMULAS FOR ∫ SINN X DX AND ∫ COSN X DX
    2. 8.2 - REDUCTION FORMULA FOR ∫ SINM X COSN X DX
    3. 8.3 - REDUCTION FORMULAS FOR ∫ TANN X DX AND ∫ SECN X DX
    4. 8.4 - REDUCTION FORMULAS FOR ∫ XN SINMX DX AND ∫ XN COSMX DX
    5. 8.5 - REDUCTION FORMULAS FOR ∫ EAX AND ∫ XM (LOG X)N DX
    6. 8.6 - REDUCTION FORMULA FOR IMN = ∫ COSM X SIN NX DX.
    7. 8.7 - REDUCTION FORMULA FOR ∫
    8. EXERCISES
  15. 9 - Quadrature and Rectification
    1. 9.1 - QUADRATURE
    2. 9.2 - RECTIFICATION
    3. EXERCISES
  16. 10 - Centre of Gravity and Moment of Inertia
    1. 10.1 - CENTRE OF GRAVITY
    2. 10.2 - MOMENT OF INERTIA
    3. 10.3 - MEAN VALUES OF A FUNCTION
    4. EXERCISES
  17. 11 - Volumes and Surfaces of Solids of Revolution
    1. 11.1 - VOLUME OF THE SOLID OF REVOLUTION (CARTESIAN EQUATIONS)
    2. 11.2 - VOLUME OF THE SOLID OF REVOLUTION (PARAMETRIC EQUATIONS)
    3. 11.3 - VOLUME OF THE SOLID OF REVOLUTION (POLAR CURVES)
    4. 11.4 - SURFACE OF THE SOLID OF REVOLUTION (CARTESIAN EQUATIONS)
    5. 11.5 - SURFACE OF THE SOLID OF REVOLUTION (PARAMETRIC EQUATIONS)
    6. 11.6 - SURFACE OF THE SOLID OF REVOLUTION (POLAR CURVES)
    7. EXERCISES
  18. 12 - Multiple Integrals
    1. 12.1 - DOUBLE INTEGRALS
    2. 12.2 - PROPERTIES OF A DOUBLE INTEGRAL
    3. 12.3 - EVALUATION OF DOUBLE INTEGRALS (CARTESIAN COORDINATES)
    4. 12.4 - EVALUATION OF DOUBLE INTEGRALS (POLAR COORDINATES)
    5. 12.5 - CHANGE OF VARIABLES IN A DOUBLE INTEGRAL
    6. 12.6 - CHANGE OF ORDER OF INTEGRATION
    7. 12.7 - AREA ENCLOSED BY PLANE CURVES (CARTESIAN AND POLAR COORDINATES)
    8. 12.8 - VOLUME AND SURFACE AREA AS DOUBLE INTEGRALS
    9. 12.9 - TRIPLE INTEGRALS AND THEIR EVALUATION
    10. 12.10 - CHANGE TO SPHERICAL POLAR COORDINATES FROM CARTESIAN COORDINATES IN A TRIPLE INTEGRAL
    11. 12.11 - VOLUME AS A TRIPLE INTEGRAL
    12. 12.12 - MISCELLANEOUS EXAMPLES
    13. EXERCISES
  19. 13 - Vector Calculus
    1. 13.1 - DIFFERENTIATION OF A VECTOR
    2. 13.2 - PARTIAL DERIVATIVES OF A VECTOR FUNCTION
    3. 13.3 - GRADIENT OF A SCALAR FIELD
    4. 13.4 - GEOMETRICAL INTERPRETATION OF A GRADIENT
    5. 13.5 - PROPERTIES OF A GRADIENT
    6. 13.6 - DIRECTIONAL DERIVATIVES
    7. 13.7 - DIVERGENCE OF A VECTOR-POINT FUNCTION
    8. 13.8 - PHYSICAL INTERPRETATION OF DIVERGENCE
    9. 13.9 - CURL OF A VECTOR-POINT FUNCTION
    10. 13.10 - PHYSICAL INTERPRETATION OF CURL
    11. 13.11 - THE LAPLACIAN OPERATOR
    12. 13.12 - PROPERTIES OF DIVERGENCE AND CURL
    13. 13.13 - INTEGRATION OF VECTOR FUNCTIONS
    14. 13.14 - LINE INTEGRAL
    15. 13.15 - WORK DONE BY A FORCE
    16. 13.16 - SURFACE INTEGRAL
    17. 13.17 - VOLUME INTEGRAL
    18. 13.18 - GAUSS'S DIVERGENCE THEOREM
    19. 13.19 - GREEN'S THEOREM IN A PLANE
    20. 13.20 - STOKE'S THEOREM
    21. 13.21 - MISCELLANEOUS EXAMPLES
    22. EXERCISES
  20. 14 - Three-Dimensional Geometry
    1. 14.1 - COORDINATE PLANES
    2. 14.2 - DISTANCE BETWEEN TWO POINTS
    3. 14.3 - DIRECTION RATIOS AND DIRECTION COSINES OF A LINE
    4. 14.4 - SECTION FORMULAE—INTERNAL DIVISION OF A LINE BY A POINT ON THE LINE
    5. 14.5 - STRAIGHT LINE IN THREE DIMENSIONS
    6. 14.6 - ANGLE BETWEEN TWO LINES
    7. 14.7 - SHORTEST DISTANCE BETWEEN TWO SKEW LINES
    8. 14.8 - EQUATION OF A PLANE
    9. 14.9 - EQUATION OF A PLANE PASSING THROUGH A GIVEN POINT AND PERPENDICULAR TO A GIVEN DIRECTION
    10. 14.10 - EQUATION OF A PLANE PASSING THROUGH THREE POINTS
    11. 14.11 - EQUATION OF A PLANE PASSING THROUGH A POINT AND PARALLEL TO TWO GIVEN VECTORS
    12. 14.12 - EQUATION OF A PLANE PASSING THROUGH TWO POINT AND PARALLEL TO A LINE
    13. 14.13 - ANGLE BETWEEN TWO PLANES
    14. 14.14 - ANGLE BETWEEN A LINE AND A PLANE
    15. 14.15 - PERPENDICULAR DISTANCE OF A POINT FROM A PLANE
    16. 14.16 - PLANES BISECTING THE ANGLES BETWEEN TWO PLANES
    17. 14.17 - INTERSECTION OF PLANES
    18. 14.18 - PLANES PASSING THROUGH THE INTERSECTION OF TWO GIVEN PLANES
    19. 14.19 - SPHERE
    20. 14.20 - EQUATION OF A SPHERE WHOSE DIAMETER IS THE LINE JOINING TWO GIVEN POINTS
    21. 14.21 - EQUATION OF A SPHERE PASSING THROUGH FOUR POINTS
    22. 14.22 - EQUATION OF THE TANGENT PLANE TO A SPHEREM
    23. 14.23 - CONDITION OF TANGENCY
    24. 14.24 - ANGLE OF INTERSECTION OF TWO SPHERES
    25. 14.25 - CONDITION OF ORTHOGONALITY OF TWO SPHERES
    26. 14.26 - CYLINDER
    27. 14.27 - EQUATION OF A CYLINDER WITH GIVEN AXIS AND GUIDING CURVES
    28. 14.28 - RIGHT CIRCULAR CYLINDER
    29. 14.29 - CONE
    30. 14.30 - EQUATION OF A CONE WITH ITS VERTEX AT THE ORIGIN
    31. 14.31 - EQUATION OF A CONE WITH GIVEN VERTEX AND GUIDING CURVE
    32. 14.32 - RIGHT CIRCULAR CONE
    33. 14.33 - RIGHT CIRCULAR CONE WITH VERTEX (α, β, γ), SEMI-VERTICAL ANGLE θ, AND THE DIRECTION COSINES OF THE AXIS.
    34. 14.34 - CONICOIDS
    35. 14.35 - SHAPE OF AN ELLIPSOID
    36. 14.36 - SHAPE OF THE HYPERBOLOID OF ONE SHEET
    37. 14.37 - SHAPE OF THE HYPERBOLOID OF TWO SHEETS
    38. 14.38 - SHAPE OF THE ELLIPTIC CONE
    39. 14.39 - INTERSECTION OF A CONICOID AND A LINE
    40. 14.40 - TANGENT PLANE AT A POINT OF CENTRAL CONICOID
    41. 14.41 - CONDITION OF TANGENCY
    42. 14.42 - EQUATION OF NORMAL TO THE CENTRAL CONICOID AT ANY POINT (α, β, γ) ON IT
    43. 14.43 - MISCELLANEOUS EXAMPLES
    44. EXERCISES
  21. 15 - Logic
    1. 15.1 - PROPOSITIONS
    2. 15.2 - BASIC LOGICAL OPERATIONS
    3. 15.3 - LOGICAL EQUIVALENCE INVOLVING TAUTOLOGIES AND CONTRADICTIONS
    4. 15.4 - CONDITIONAL PROPOSITIONS
    5. EXERCISES
  22. 16 - Elements of Fuzzy Logic
    1. 16.1 - FUZZY SET
    2. 16.2 - STANDARD OPERATIONS ON A FUZZY SET
    3. 16.3 - MANY VALUED LOGIC
    4. 16.4 - FUZZY LOGIC
    5. 16.5 - FUZZY PROPOSITIONS
    6. EXERCISES
  23. 17 - Graphs
    1. 17.1 - DEFINITIONS AND BASIC CONCEPTS
    2. 17.2 - SPECIAL GRAPHS
    3. 17.3 - SUBGRAPHS
    4. 17.4 - ISOMORPHISMS OF GRAPHS
    5. 17.5 - WALKS, PATHS AND CIRCUITS
    6. 17.6 - EULERIAN PATHS AND CIRCUITS
    7. 17.7 - HAMILTONIAN CIRCUITS
    8. 17.8 - MATRIX REPRESENTATION OF GRAPHS
    9. 17.9 - PLANAR GRAPHS
    10. 17.10 - COLOURING OF GRAPH
    11. 17.11 - DIRECTED GRAPHS
    12. 17.12 - TREES
    13. 17.13 - ISOMORPHISM OF TREES
    14. 17.14 - REPRESENTATION OF ALGEBRAIC EXPRESSIONS BY BINARY TREES
    15. 17.15 - SPANNING TREE OF A GRAPH
    16. 17.16 - SHORTEST PATH PROBLEM
    17. 17.17 - MINIMAL SPANNING TREE
    18. 17.18 - CUT SETS
    19. 17.19 - TREE SEARCHING
    20. 17.20 - TRANSPORT NETWORKS
    21. EXERCISES
  24. Copyright