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. Linear Permutations Without Repeatition
   3. Linear Permutations With Unrestricted Repeatition
   4. Linear Permutations With Restricted Repeatition
   5. Circular Permutations Without Repeatition
   6. Circular Permutations With Unrestricted Repeatition
   7. Combinations Without Repeatition
   8. Combinations With Unrestricted Repeatition
   9. Using Generating Functions to Find Combinations, Count of Combinations and Count of Linear Permutations
   10. Permutation Tables, Permutation Cycles and Transpositions
   11. Decomposition of Permutation/Permutation Cycles into Transpositions
   12. Product of Permutations, Permutation Cycles and Transpositions
   13. Inverse and Order of a Permutation
   14. Permutations and Permutation Matrices
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
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. Vector Types and their Diagramatic / Visual / Symbolic Representation
      7. Laws of Vector Addition/Subtraction
      8. Dot/Scalar/Inner Product of Vectors in Arbitrary Non Standard Basis
      9. Geometric Interpretation of Dot/Inner Product
      10. Covariant and Contravariant Components of a Vector
      11. Cross Product of Vectors
      12. Geometric Interpretation of Cross Product
      13. Cross Product of Vectors in Arbitrary Non Standard Basis
      14. Scalar Triple Product
      15. Vector Triple Product
      16. Scalar Quad Product
      17. Vector Quad Product
      18. Orthogonal Vector Projection/Rejection
      19. Non-Orthogonal/Oblique Vector Projection/Rejection
      20. Determinant Product of Vectors
      21. Determinant Product of Vectors in Arbitrary Non Standard Basis
      22. Wedge Product of Vectors
      23. Wedge Product of Vectors in Arbitrary Non Standard Basis
      24. Dyads and Dyadics
      25. Dot Product between a Dyad and a Vector
      26. Cross Product between a Dyad and a Vector
      27. Dot Product between 2 Dyads/Dyadics
      28. Double-Dot Product between 2 Dyads/Dyadics
      29. Dot-Cross Product between 2 Dyads/Dyadics
      30. Cross-Dot Product between 2 Dyads/Dyadics
      31. 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. 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. 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
       6. 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
       7. Circle and Imaginary Circle
       8. 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
       9. Pair of Lines
       10. Point(s) of Intersection Between a Line and a Conic
       11. Point(s) of Intersection Between 2 Conics
   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. Ellipsoid and Imaginary Ellipsoid
       5. Sphere and Imaginary Sphere
       6. Elliptical/Circular Cone and Imaginary Elliptical/Circular Cone
       7. Elliptical/Circular Cylinder and Imaginary Elliptical/Circular Cylinder
       8. Parabolic Cylinder
       9. Hyperbolic Cylinder
       10. Elliptic Paraboloid
       11. Hyperbolic Paraboloid
       12. Hyperboloid of 1 Sheet
       13. Hyperboloid of 2 Sheets
       14. 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