Mathematics





Mathematics 

Algebra

Euclidean And Analytical Geometry

Probability


Linear Algebra



Linear Algebra

 Introduction to Matrices - Part I   Introduction to Matrices. Theory, definitions. What a Matrix is, order of a matrix, equality of matrices, different kind of matrices: row matrix, column matrix, square matrix, diagonal, identity and triangular matrices. Definitions of Trace, Minor, Cofactors, Adjoint, Inverse, Transpose of a matrix. Addition, subtraction, scalar multiplication, multiplication of matrices. Defining special types of matrices like Symmetric, Skew Symmetric, Idempotent, Involuntary, Nil-potent, Singular, Non-Singular, Unitary matrices.


Linear Algebra
Introduction to Matrices - Part II Problems and solved examples based on the sub-topics mentioned above. Some of the problems in this part demonstrate finding the rank, inverse or characteristic equations of matrices. Representing real life problems in matrix form.



Linear Algebra
Determinants Introduction to determinants. Second and third order determinants, minors and co-factors. Properties of determinants and how it remains altered or unaltered based on simple transformations is matrices. Expanding the determinant. Solved problems related to determinants. 
 
Linear Algebra
Simultaneous linear equations in multiple variables Representing a system of linear equations in multiple variables in matrix form. Using determinants to solve these systems of equations. Meaning of consistent, homogeneous and non-homogeneous systems of equations. Theorems relating to consistency of systems of equations. Application of Cramer rule. Solved problems demonstrating how to solve linear equations using matrix and determinant related methods.





Linear Algebra
Basic concepts in Linear Algebra and Vector spaces Theory and definitions. Closure, commutative, associative, distributive laws. Defining Vector space, subspaces, linear dependence, dimension and bias. A few introductory problems proving certain sets to be vector spaces. 
Linear Algebra
Introductory problems related to Vector Spaces - Problems demonstrating the concepts introduced in the previous tutorial. Checking or proving something to be a sub-space, demonstrating that something is not a sub-space of something else, verifying linear independence; problems relating to dimension and basis; inverting matrices and echelon matrices.




Linear Algebra
More concepts related to Vector Spaces Defining and explaining the norm of a vector, inner product, Graham-Schmidt process, co-ordinate vectors, linear transformation and its kernel. Introductory problems related to these.



Linear Algebra
Problems related to linear transformation, linear maps and operators - Solved examples and problems related to linear transformation, linear maps and operators and other concepts discussed theoretically in the previous tutorial. 

Linear Algebra

Definitions of Rank, Eigen Values, Eigen Vectors, Cayley Hamilton Theorem 
Eigenvalues, eigenvectors, Cayley Hamilton Theorem





Linear Algebra

More Problems related to Simultaneous Equations; problems related to eigenvalues and eigenvectors  Demonstrating the Crammer rule, using eigenvalue methods to solve vector space problems, verifying Cayley Hamilton Theorem, advanced problems related to systems of equations. Solving a system of differential equations .





 
Linear Algebra

A few closing problems in Linear Algebra Solving a recurrence relation, some more of system of equations.




 

Vectors

Vectors1a
Vectors 1a ( Theory and Definitions: Introduction to Vectors; Vector, Scalar and Triple Products)
Introducing a vector, position vectors, direction cosines, different types of vectors, addition and subtraction of vectors. Vector and Scalar products. Scalar Triple product and Vector triple product and their properties. Components and projections of vectors.

Vectors1b
Vectors 1b ( Solved Problem Sets: Introduction to Vectors; Vector, Scalar and Triple Products )
Solved examples and problem sets based on the above concepts.

vec2a
Vectors 2a ( Theory and Definitions: Vectors and Geometry ) Vectors and geometry. Parametric vectorial equations of lines and planes. Angles between lines and planes. Co-planar and collinear points. Cartesian equations for lines and planes in 3D.

vectors
Vectors 2b ( Solved Problem Sets: Vectors and Geometry )

Solved examples and problem sets based on the above concepts.



Vectors
 
 
Vectors 3a ( Theory and Definitions: Vector Differential and Integral Calculus ) Vector Differential Calculus. Derivative, curves, tangential vectors, vector functions, gradient, directional derivative, divergence and curl of a vector function; important formulas related to div, curl and grad. Vector Integral Calculus. Line integral, independence of path, Green's theorem, divergence theorem of Gauss, green's formulas, Stoke's theorems.


Vectors 3b
Vectors 3b ( Solved Problem Sets: Vector Differential and Integral Calculus ) - Solved examples and problem sets based on the above concepts.

 


Trigonometry


Trigonometry 1a ( Introduction to Trigonometry - Definitions, Formulas ) Introducing trigonometric ratios, plots of trigonometric functions, compound angle formulas. Domains and ranges of trigonometric functions, monotonicity of trigonometric functions quadrant wise. Formulas for double and triple angle ratios.

Inverse Trigonometric Ratios


Trigonometry 1b ( Tutorial with solved problems based on Trigonometric ratios ) Problems based on the concepts introduced above.



Trigonometry 2a ( Basic concepts related to Heights and Distances ) Applying trigonometry to problems involving heights and distances. Angles of elevation and depression. Sine and Cosine rule, half angle formulas. Circumradius, inradius and escribed radius. Circumcentre, incentre, centroid and median of a triangle.

Trigonometric 2b


Trigonometry 2b ( Tutorial with solved problems related to Heights and Distances and other applications of Trigonometry ) - Problems based on the concepts introduced above.



Trigonometry 3a ( Introducing Inverse Trigonometric Ratios)
Inverse trigonometric ratios - their domains, ranges and plots.



 
Inverse Trigonometric Ratios

Trigonometry 3b ( Tutorial with solved problems related to inverse trigonometric ratios )- Problems related to inverse trigonometric ratios.

Trigonometric Equations

Trigonometry 4 ( A tutorial on solving trigonometric equations )- Solving trigonometric equations. Methods and transformations frequently used in solving such equations.

 



Single Variable Calculus


Continutiy and differentiability

 Quick and introductory definitions related to Funtions, Limits and Continuity - Defining the domain and range of a function, the meaning of continuity, limits, left and right hand limits, properties of limits and the "lim" operator; some common limits;  defining the L'Hospital rule, intermediate and extreme value theorems.  

Domain and Range

 

 

Functions, Limits and Continuity - Solved Problem Set I - The Domain, Range, Plots and Graphs of Functions;  L'Hospital's Rule- -  Solved problems demonstrating how to compute the domain and range of functions, drawing the graphs of functions, the mod function, deciding if a function is invertible or not; calculating limits for some elementary examples, solving 0/0 forms, applying L'Hospital rule.

Domain and Range

 Functions, Limits and Continuity - Solved Problem Set II - Conditions for Continuity, More Limits, Approximations for ln (1+x) and sin x for infinitesimal values of x  

More advanced cases of evaluating limits, conditions for continuity of functions, common approximations used while evaluating limits for ln ( 1 + x ), sin (x); continuity related problems for more advanced functions than the ones in the first group of problems (in the last tutorial).

Domain and Range

 Functions, Limits and Continuity - Solved Problem Set III - Continuity and Intermediate Value Theorems - Problems related to Continuity, intermediate value theorem.

differentiation

 

Introductory concepts and definitions related to Differentiation - Basic formulas, Successive Differentiation, Leibnitz, Rolle and Lagrange Theorems, Maxima , Minima, Convexity, Concavity, etc - Theory and definitions introducing differentiability, basic differentiation formulas of common algebraic and trigonometric functions , successive differentiation, Leibnitz Theorem, Rolle's Theorem,  Lagrange's Mean Value Theorem, Increasing and decreasing functions, Maxima and Minima; Concavity, convexity and inflexion, implicit differentiation. 

differentiation

Differential Calculus - Solved Problem Set I - Common Exponential, Log , trigonometric and polynomial functions  - Examples and solved problems - differentiation of common algebraic, exponential, logarithmic, trigonometric and polynomial functions and terms; problems related to differentiability . 

  Differentiation- Continuity; Changing Independent Variables

Differential Calculus - Solved Problem Set II - Derivability and continuity of functins - Change of Indepndent Variables - Finding N-th Derivatives -
 Examples and solved problems - related to derivability and continuity of functions; changing the independent variable in a differential equation; finding the N-th derivative of functions

  Differentiation- Continuity; Changing Independent Variables

Differential Calculus - Solved Problems Set III- Maximia, Minima, Extreme Values,  Rolle's Theorem - 
Examples and solved problems - related to increasing and decreasing functions; maxima, minima and extreme values; Rolle's Theorem 

Differentiation- Curves

Differential Calculus - Solved Problems Set IV - Points of Inflexion, Radius of Curvature, Curve Sketching -  
Examples and solved problems - Slope of tangents to a curve, points of inflexion, convexity and concavity of curves, radius of curvature and asymptotes of curves, sketching curves 

Differentiation- Curves

Differential Calculus - Solved Problems Set V - Curve Sketching, Parametric Curves - More examples of investigating and sketching curves, parametric representation of curves

Integral Calculus- Introducing Definite and Indefinite Integrals

Introducing Integral Calculus - Definite and Indefinite Integrals - using Substitution , Integration By Parts, ILATE rule  - Theory and definitions. What integration means, the integral and the integrand. Indefinite integrals, integrals of common functions.  Definite integration and properties of definite integrals; Integration by  substitution, integration by parts, the LIATE rule, Integral as the limit of a sum. Important forms encountered in integration. 

Integral Calculus- Introducing Definite and Indefinite Integrals

Integral Calculus - Solved Problems Set I - Basic examples of polynomials and trigonometric functions, area under curves - Examples and solved problems - elementary examples of integration involving trigonometric functions, polynomials; integration by parts; area under curves.

 Integral Calculus- Introducing Definite and Indefinite Integrals

Integral Calculus - Solved Problems Set II - More integrals, functions involving trigonometric and inverse trigonometric ratios - Examples and solved problems - integration by substitution, definite integrals, integration involving trigonometric and inverse trigonometric ratios.

Integral Calculus- Introducing Definite and Indefinite Integrals

Integral Calculus - Solved Problems Set III - Reduction Formulas, Using Partial FractionsIExamples and solved problems - Reduction formulas, reducing the integrand to partial fractions, more of definite integrals

Integral Calculus- Introducing Definite and Indefinite Integrals

Integral Calculus - Solved Problems Set IV - More of integration using partial fractions, more complex substitutions and transformations - Examples and solved problems - More of integrals involving partial fractions, more complex substitutions and transformations

Integral Calculus- Introducing Definite and Indefinite Integrals

Integral Calculus - Solved Problems Set V- Integration as a summation of a series - Examples and solved problems - More complex examples of integration, examples of integration as the limit of a summation of a series 

Differential Equations

Introduction to Differential Equations and Solved Problems - Set I - Order and Degree, Linear and Non-Linear Differential Equations, Homogeneous Equations, Integrating Factor - 
 Theory and definitions. What a differential equation is; ordinary and partial differential equations; order and degree of a differential equation; linear and non linear differential equations; General, particular and singular solutions; Initial and boundary value problems; Linear independence and dependence; Homogeneous equations; First order differential equations; Characteristic and auxiliary equations. Introductory problems demonstrating these concepts. Introducing the concept of Integrating Factor (IF). 

Differential Equations

Differential Equations - Solved Problems - Set II - D operator, auxillary equation, General Solution - Examples and solved problems - Solving linear differential equations, the D operator, auxiliary equations. Finding the general solution ( CF + PI )

Differential Equations

Differential Equations - Solved Problems - Set III - More Differential Equations - More complex cases of differential equations. 

Differential Equations

Differential Equations - Solved Problems - Set IV -
 Still more differential equations. 



Multiple Variable Calculus

Applied Mathematics : An Introduction to Game Theory

Applied Mathematics : An Introduction to Operations Research

Operations Research

 Introduction to Operations Research
A quick introduction to Operations Research. Introducing Linear Programming, standard and canonical forms. Linear Programming geometry, feasible regions, feasible solutions, simplex method. Some basic problems.

 

Mathematics & Computing - A few programs in C


Basic Number Theory

                    C Program: Source code to solve the Josephus Problem
               C Program: Using the Sieve of Eratosthenes to print Prime Numbers
               C Program: Check for Armstrong Numbers
               C Program: Source Code for computing the GCD(HFC) of two numbers

Linear Algebra and Matrices

       C Program: Computing the Upper Triangular Matrix and Lower Triangular Matrix
     C Program: Demonstrating Operations on Matrices - Addition, Subtraction, Multiplication, Inversion, Finding Determinants
     C Program: Solving Simultaneous Equations in Two Variables
     C Program: Source Code for Solving Quadratic Equations

Popular Examples from Numerical Computing





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Prashant Bhattacharji,
Jun 30, 2011, 6:21 AM
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