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Maths: Table of Contents

  1. Logarithms and Anti Logarithms
    1. Logarithm Formulas
    2. Growth, Reduction, Simple and Compound Interest and the Value of 'e'
  2. Set Theory
    1. Rules of Operation on Sets
    2. List of Frequently Referenced Sets in Mathematics
  3. Group Theory
    1. Cayley Tables: Structural Representation of Finite Groups
  4. Combinatorics
    1. Fundamental Principle of Counting, Concept of Factorial, Permutation and Combination
    2. Symmetry Independent Permutations Without Repeatition
    3. Combinations Without Repeatition
    4. Linear Symmetry Permutations Without Repeatition
    5. Circular Symmetry Permutations Without Repeatition
    6. Necklace Symmetry Permutations Without Repeatition
    7. Permutation Tables, Permutation Cycles and Transpositions
    8. Cycle Index Count of a Permutation
    9. Decomposition of Permutation/Permutation Cycles into Transpositions
    10. Product of Permutations, Permutation Cycles and Transpositions
    11. Inverse and Order of a Permutation
    12. Permutations and Permutation Matrices
    13. Combinations With Repeatition
    14. Symmetry Independent Permutations With Repeatition
    15. Using Generating Functions to Find Combinations With Repeatitions, Count of Combinations With Repeatitions and Count of Linear Permutations With Repeatitions
  5. Trigonometry
    1. Adding/Subtracting/Negating Angles of Trignometric Ratios
    2. Law of Sines for Triangles on Plane Surface
    3. Law of Cosines for Triangles on Plane Surface
    4. Time Period, Frequency and Phase of Trigonometric Sine and Cosine Functions
    5. Trignometric/Hyperbolic Identities
  6. Polynomial Expressions/Equations and Rational Expressions
    1. General Polynomial Equations
    2. Finding Roots of a Quadratic Polynomial Equation
    3. Finding Roots of a Cubic Polynomial Equation
    4. Finding Roots of a Quartic Polynomial Equation
    5. Finding Roots of a Polynomial Equation of Any Arbitrary Degree
  7. Matrix, Vector and Tensor Algebra
    1. Matrix Algebra
      1. Matrix Vectorization
      2. Trace of a Square Matrix
      3. Transpose/Conjugate Transpose of a Matrix
      4. Element Wise Matrix Addition/Subtraction and NULL Matrix
      5. Direct Sum of Matrices
      6. Matrix Multiplication with a Scalar
      7. Dot Product of 2 Row/Column Matrices
      8. Matrix Multiplication: Inner Product of Matrices
      9. Hadamard Product: Element Wise Matrix Multiplication
      10. Double-Dot Product of 2 Matrices
      11. Kronecker Product: Outer Product of Matrices
      12. Tensor Product of Matrices
      13. Symmetric and Skew Symmetric Matrices
      14. Hermitian and Anti Hermitian Matrices
      15. Triangular and Trapezoidal Matrices
      16. Identity, Scalar and Diagonal Matrices
      17. Periodic, Indempotent, Involutary and Nilpotent Matrices
      18. Orthogonal and Unitary Matrices
      19. Permutation Matrices
      20. Determinant, Minor, Cofactor and Adjoint of a Square Matrix
      21. Principal Minors and Traces of Principal Minors of a Square Matrix
      22. Row Echelon and Column Echelon Matrix
      23. Elementary Row/Column Operations on a Matrix
      24. Matrices and System of Linear Equations
      25. Solving System of Linear Equations Using Row Operations/Gaussian Elimination
      26. Column Space, Row Space, NULL Space and Orthogonal Space of a Matrix
      27. Linear Dependence/Independence of Vectors in a Matrix and Rank of a Matrix
      28. Vector Space of a Matrix and Rank of a Matrix
      29. Solving System of Linear Equations Using Cramer's Rule
      30. Inverse of a Square Matrix
      31. Solving System of Linear Equations Using Inverse of Matrix
      32. Gramian Matrix / Gram Matrix / Metric Tensor
      33. Dual of a Vector/Matrix
      34. Basis Vector Matrix and Vector Space / Subspace Spanned by a Basis Vector Matrix
      35. Projection/Rejection Matrices and Projected/Rejected Vectors
      36. Matrix Factorization through LU / PLU / LUP Decomposition using Elementary Row/Column Operations
      37. Matrix Factorization through QR Decomposition using Gram-Schmidt Process
      38. Characteristic Polynomial / Polynomial Equation of a Square Matrix
      39. Eigen-Values and Eigen-Vectors of a Square Matrix
      40. Matrix Factorization through Eigen-Value / Eigen-Vector Decomposition
      41. Matrix Factorization through Singular Value Decomposition
    2. Vector Algebra
      1. Matrix Representation of Vectors
      2. Concept of Basis Vectors and Directional Representation of Vectors
      3. Change of Basis Vectors for a Vector
      4. Basis Vector Transformation
      5. Dot/Scalar/Inner Product of Vectors, Magnitude of Vectors and Unit Vectors
      6. Dot/Scalar/Inner Product of Vectors in Arbitrary Non Standard Basis
      7. Geometric Interpretation of Dot/Scalar/Inner Product of Real Vectors
      8. Vector Types and their Diagramatic / Visual / Symbolic Representation
      9. Laws of Addition/Subtraction of Real Vectors
      10. Covariant and Contravariant Components of a Vector
      11. Cross Product of Vectors
      12. Cross Product of Vectors in Arbitrary Non Standard Basis
      13. Geometric Interpretation of Cross Product of Real Vectors
      14. Determinant Product of Vectors
      15. Determinant Product of Vectors in Arbitrary Non Standard Basis
      16. Wedge Product of Vectors
      17. Wedge Product of Vectors in Arbitrary Non Standard Basis
      18. Scalar Triple Product
      19. Vector Triple Product
      20. Scalar Quad Product
      21. Vector Quad Product
      22. Orthogonal Vector Projection/Rejection
      23. Non-Orthogonal/Oblique Vector Projection/Rejection
    3. Dyads and Dyadics Algebra
      1. Dot Product between a Dyad and a Vector
      2. Cross Product between a Dyad and a Vector
      3. Dot Product between 2 Dyads/Dyadics
      4. Double-Dot Product between 2 Dyads/Dyadics
      5. Dot-Cross Product between 2 Dyads/Dyadics
      6. Cross-Dot Product between 2 Dyads/Dyadics
      7. Double-Cross Product between 2 Dyads/Dyadics
  8. Coordinate Systems and Coordinate Geometry
    1. Cartesian Coordinate Systems
      1. Distance Formula
      2. Section Formula
      3. Finding Polar and Equatorial Angles of a Point
      4. Rules for Measurement of Rotation Angles on a Plane
    2. Curvilinear Coordinate Systems
    3. Polar Coordinate System
    4. Parabolic Coordinate System
    5. Spherical Coordinate System
    6. Polar Cylindrical Coordinate System
    7. Parabolic Cylindrical Coordinate System
    8. Representing Geometric Objects/Fields in Coordinate Systems
    9. Transformation
      1. Translation
      2. Rotation
        1. Derivation of Rodrigues Rotation Formula and Formula for 3D Rotation Matrix
        2. Calculating 3D Rotation Matrix using Polar and Equatorial Angles of Axis of Rotation
        3. Calculating 3D Rotation Matrix using Euler and Tait-Bryan Angles/Matrices
        4. Finding Axes and Angles of Rotation from Rotation Matrix
        5. Finding Euler/Tait Bryan Angles from Rotation Matrix
        6. Improper Rotations and Roto-Reflection Matrices in 3 Dimensions
        7. Factoring a 3D Roto-Reflection Matrix into a Rotation and Reflection Matrix
        8. Rotation Axes (Yaw, Pitch, Roll) and Orientation (Heading, Attitude, Bank)
      3. Reflection
        1. Derivation of Reflection Formula Across a Line
        2. Derivation of Reflection Formula Across a Line in 2D/Plane in 3D/Hyper-Plane in Higher Dimensions
      4. Scaling
      5. Shearing
    10. Linear Equations, Lines, Planes and Hyper-Planes
      1. Lines
        1. Derivation/Representation of Equation of Lines
        2. Finding Points on Line/Intercepts of Line
        3. Types of Lines in 2D
        4. Types of Lines in 3D
        5. Condition for Collinearity of 3 Points
        6. Angular Slope of a Line in 2D
        7. Angular Normal of a Line in 2D
        8. Angle Between 2 Lines
        9. Relation Between 2 Lines
        10. Condition for Concurrency of Lines
        11. Family of Lines in 2D
      2. Planes
        1. Derivation/Representation of Equation of Planes
        2. Finding Points on Plane/Intercepts of Plane
        3. Types of Planes
        4. Condition for Coplanarity of 4 Points
        5. Projection of Vector on a Plane
        6. Angular Normal of a Plane
        7. Angle Between 2 Planes
        8. Angle Between a Line and a Plane
        9. Relation Between a Line and a Plane
        10. Relation Between 2 Planes
        11. Relation Between 3 Planes
        12. Condition for Collinearity and Concurrency of Planes
        13. Family of Planes
      3. Distance of Point from a Line/Plane/Hyper-Plane
      4. Projection of Point on a Line/Plane/Hyper-Plane
    11. General Quadratic Equations in 2 Variables and Conic Sections
      1. Conic Section Translation
      2. Conic Section Rotation
      3. Conic Section Normalization
      4. Centers of Central Conic Section Curves
      5. Point(s) of Intersection Between a Line and a Conic
      6. Point(s) of Intersection Between 2 Conics
      7. Projection of a Point on Conic and Distance of a Point from Conic Using Conic Intersection
      8. Projection of a Point on Conic and Distance of a Point from Conic Using Normal to the Conic
      9. Parabola
        1. Derivation of Standard and Explicit Coordinate Equations for Axis Aligned Parabolas
        2. Finding Parameters of Axis Aligned Parabola from Standard Coordinate Equation
        3. Finding Parameters of Axis Aligned Parabola from Explicit Coordinate Equation
        4. Derivation and Properties of Implicit Coordinate Equation for Axis Aligned and Arbitrarily Rotated Parabolas
        5. Finding Parameters of Axis Aligned Parabola from Implicit Coordinate Equation
        6. Finding Parameters of Arbitrarily Rotated Parabola from Implicit Coordinate Equation
        7. Finding Equation of Axis Aligned Parabolas from given Focal Length and Vertex
        8. Finding Equation of Axis Aligned Parabolas from given Focal Length and Focus
        9. Finding Equation of Axis Aligned Parabolas from 3 Non-Collinear Points
        10. Finding Equation of Parabola from given Focus and Directrix
        11. Finding Equation of Parabola from given Focus and Vertex
        12. Finding Equation of Parabola from given Focus and Base
        13. Finding Equation of Parabola from given Vertex and Directrix
        14. Finding Equation of Parabola from given Vertex and Latus Rectum
        15. Parametric Equations and Position Vector Representation of Parabola
        16. Converting Parabola Equation from Standard Coordinate to Standard Parametric
        17. Converting Parabola Equation from Standard Parametric to Standard Coordinate
        18. Converting Parabola Equation from Explicit Coordinate to Parametric
        19. Converting Parabola Equation from Axis Aligned Parametric to Explicit/Implicit Coordinate
        20. Converting Parabola Equation from General Parametric to Implicit Coordinate
        21. Converting Parabola Equation from Implicit Coordinate to General Parametric
      10. Ellipse and Imaginary Ellipse
        1. Derivation of Standard and Implicit Coordinate Equation for Axis Aligned Ellipses
        2. Finding Parameters of Axis Aligned Ellipses from Standard Coordinate Equation
        3. Finding Parameters of Axis Aligned Ellipses from Implicit Coordinate Equation
        4. Derivation of Implicit Coordinate Equation for Arbitrarily Rotated and Translated Ellipses
        5. Finding Parameters of Arbitrarily Rotated and Translated Ellipse from Implicit Coordinate Equation
        6. Finding Equation of Ellipse from given 2 Foci and Major Axis Length
        7. Finding Equation of Ellipse from a given Focus, a Vertex and Eccentricity
        8. Finding Equation of Ellipse from given Directrix, Adjacent Focus and Eccentricity
        9. Finding Equation of Ellipse from given Directrix, Adjacent Vertex and Eccentricity
        10. Finding Parametric Equations for Axis Aligned and Rotated Ellipse
      11. Circle and Imaginary Circle
      12. Hyperbola
        1. Derivation of Standard and Implicit Coordinate Equation for Axis Aligned Hyperbolas
        2. Finding Parameters of Axis Aligned Hyperbolas from Standard Coordinate Equation
        3. Finding Parameters of Axis Aligned Hyperbolas from Implicit Coordinate Equation
        4. Derivation of Implicit Coordinate Equation for Arbitrarily Rotated and Translated Hyperbolas
        5. Finding Parameters of Arbitrarily Rotated and Translated Hyperbola from Implicit Coordinate Equation
        6. Conjugate Hyperbola
        7. Asymptotic Lines of Hyperbola and Conjugate Hyperbola
        8. Rectangular Hyperbola
        9. Finding Equation of Hyperbola from given 2 Foci and Transverse Axis Length
        10. Finding Equation of Hyperbola from a given Focus, a Vertex and Eccentricity
        11. Finding Equation of Hyperbola from given Directrix, Adjacent Focus and Eccentricity
        12. Finding Equation of Hyperbola from given Directrix, Adjacent Vertex and Eccentricity
        13. Finding Parametric Equations for Axis Aligned and Rotated Hyperbola Based on Secant and Tangent Ratios
        14. Finding Parametric Equations for Axis Aligned and Rotated Hyperbola Based on Hypebolic Sine and Cosine
      13. Pair of Lines
    12. General Quadratic Equations in 3 Variables and Quadric Surfaces
      1. Quadric Surface Translation
      2. Quadric Surface Rotation
      3. Centers of Central Quadric Surfaces
      4. Projection of a Point on Quadric Surface and Distance of a Point from Quadric Surface Using Normal to the Quadric Surface
      5. Ellipsoid and Imaginary Ellipsoid
      6. Sphere and Imaginary Sphere
      7. Elliptical/Circular Cone and Imaginary Elliptical/Circular Cone
      8. Elliptical/Circular Cylinder and Imaginary Elliptical/Circular Cylinder
      9. Parabolic Cylinder
      10. Hyperbolic Cylinder
      11. Elliptic Paraboloid
      12. Hyperbolic Paraboloid
      13. Hyperboloid of 1 Sheet
      14. Hyperboloid of 2 Sheets
      15. Pair of Planes
  9. Differential Calculus and Geometry
    1. Rules for Calculating Derivatives and Differentials for Functions of a Single Variable
    2. Rules for Calculating Derivatives and Differentials for Functions of Multiple Variables
    3. Gradient Vector, Jacobian Matrix and Hessian Matrix
    4. Representation of Curves
    5. Tangent Vector to a Curve and Arc Length / Differential Arc Length of a Curve
    6. Curvature and Torsion of a Curve
    7. Frenet-Serret Frame of a Curve and Derivation of Frenet-Serret Equations
    8. Order of Contact Between 2 Curves and the Concept of Osculation
    9. Osculating Circle, Radius and Center of Curvature of a Curve
    10. Center and Radius of Osculating Sphere of a Curve
    11. Vertex and Evolute of a Curve
    12. Representation of Surfaces
    13. Tangent and Normal Vectors of a Surface
  10. Integral Calculus and Geometry
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