www.thelearningpoint.net Come Check out our Free Repository of Tutorials and Visualizations ! Geometry in Nature : A Tree made with Fractals; A Complex Function

Click below if you'd like to check out our recent additions in the Physics section !

Mathematics Algebra Introduction to Complex NumbersIntroduction to Complex Numbers and iota. Argand plane and iota. Complex numbers as free vectors. N-th roots of a complex number. Notes, formulas and solved problems related to these sub-topics.

The Principle of Mathematical Induction Introductory problems related to Mathematical Induction.

Quadratic EquationsIntroducing various techniques by which quadratic equations can be solved - factorization, direct formula. Relationship between roots of a quadratic equation. Cubic and higher order equations - relationship between roots and coefficients

Quadratic Inequalities Quadratic inequalities. Using factorization and visualization based methods.

for these. Graphs and plots of quadratic equations.

Series and ProgressionsArithmetic, Geometric, Harmonic and mixed progressions. Notes, formulas and solved problems. Sum of the first N terms. Arithmetic, Geometric and Harmonic means and the relationship between them.

Geometry

Geometry

Co-ordinate Geometry

Introduction to Co-ordinate Geometry

Equation of Straight Line

The Circle A Quick Introduction to Conic Sections: Parabola, Hyperbola, Ellipse

Parabola

Hyperbola

Ellipse

Probability

Probability - Part Zero - A Very Basic Introduction

Probability - Part 1 - Basic Probability

Probability - Part 2 - A Tutorial on Probability

Probability - Part 3 - Joint Probability, Bivariate Normal Distributions,

Definitions, Random Variables

Distributions Functions of Random Variable,Transformation of Random Vectors

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.

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.

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.

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.

Basic concepts in Linear Algebra and Vector spaces Theory

Introductory problems related to Vector Spaces - Problems

More concepts related to Vector Spaces Defining and

Problems related to linear transformation, linear maps and operators -

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.

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.

explaining the norm of a vector, inner product, Graham-Schmidt process, co-ordinate vectors, linear transformation and its kernel. Introductory problems related to these.

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

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

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 .

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

Vectors Vectors 1a ( Theory and Definitions: Introduction to Vectors; Vector, Scalar and Triple Products)

Vectors 1b ( Solved Problem Sets: Introduction to Vectors; Vector, Scalar and Triple Products )Solved examples and

Vectors 2a ( Theory and Definitions: Vectors and

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

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.

problem sets based on the above concepts.

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.

problem sets based on the above concepts.

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 ( 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

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

Trigonometry 2a ( Basic concepts related to Heights and Distances ) Applying trigonometry

Trigonometry 2b ( Tutorial with solved problems related to Heights and Distances and other applications

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.

introduced above. 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.

of Trigonometry ) - Problems based on the concepts introduced above.

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

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

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

Single Variable Calculus 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.

Functions, Limits and Continuity - Solved Problem Set I - 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

Functions, Limits and Continuity - Solved Problem Set II - 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).

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

not; calculating limits for some elementary examples, solving 0/0 forms, applying L'Hospital rule.

Introductory concepts and definitions related to Differentiation - 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.

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

Differential Calculus - Solved Problem Set II - 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

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

Differential Calculus - Solved Problems Set IV - 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

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

Introducing Integral Calculus - 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

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

integration.

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

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

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

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

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

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

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

Multiple Variable Calculus

Calculus - Multiple Variables - Part I- Functions of severable variables; limits and continuity

Calculus - Multiple Variables - Part 2- Functions of several variables, theorems and co-ordinates

Calculus - Multiple Variables - Part 3- Multiple Integrals; double and triple integrals

Applied Mathematics : An Introduction to Game Theory An Introduction to Game Theory

Extensive Games Bayesian Games : Games with Incomplete Information

Repeated Games

Applied Mathematics : An Introduction to 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.

PhysicsBasic Mechanics Introduction to Vectors and Motion

Vectors and Projectile Motion

Newton's Laws of Motion

Work, Force and Energy

Simple Harmonic Motion

Rotational Dynamics

Fluid Mechanics

Engineering Mechanics

Moments and Equivalent Systems

Centroid And Center of Gravity

Analysis of Structures

Electrostatics and Electromagnetism

Electrostatics - Part 1: Theory, definitions and problems Columb's law. Electric Field Intensity, principle of superposition, gauss theorem, electrostatic potential, electric field intensities due to common charge distributions, capacitors and calculating capacitance. Solved problems.

Electrostatics - Part 2: More solved problems.More solved problems related to the concepts introduced above.

Electromagnetism - Part 1: Theory and Definitions Lorentz Force, Bio-Savart law, Ampere's force law, basic laws related to Magnetic fields and their applications. Magnetic field intensities due to common current distributions. Electromagnetic Induction. Self and mutual induction.

Electromagnetism - Part 2: Solved problemsSolved problems related to the concepts introduced above.

Advanced concepts in Electrostatics and Electromagnetism ( Theory only )Advanced concepts related to electrostatics and electromagnetism (theory only).

Computer Science and Programming

Data Structures and Algorithms Arrays : Popular Sorting and Searching Algorithms

Bubble Sort - One of the most elementary sorting algorithms to implement -

Insertion Sort - Another quadratic time sorting algorithm - an

Selection SortAnother quadratic time sorting

Shell SortAn inefficient but interesting

and also very inefficient. Runs in quadratic time. A good starting point to understand sorting in general, before moving on to more advanced techniques and algorithms. A general idea of how the algorithm works and a the code for a C program.

example of dynamic programming. An explanation and step through of how the algorithm works, as well as the source code for a C program which performs insertion sort.

algorithm - an example of a greedy algorithm. An explanation and step through of how the algorithm works, as well as the source code for a C program which performs selection sort.

algorithm, the complexity of which is not exactly known.

Merge Sort An example of a Divide and Conquer algorithm. Works in O(n log n) time. The memory complexity for this is a bit of a disadvantage.

Quick Sort In the average case, this works in O(n log n) time. No additional memory overhead - so this is better than merge sort in this regard. A partition element is selected, the array is restructured such that all elements greater or less than the partition are on opposite sides of the partition. These two parts of the array are then sorted recursively.

Heap SortEfficient sorting algorithm which runs in O(n log n) time. Uses the Heap data structure.

Binary Search AlgorithmCommonly used algorithm used to find the position of an element in a sorted array. Runs in O(log n) time.

Basic Data Structures and Operations on them

Stacks Last In First Out data structures ( LIFO ). Like a stack of cards from which you pick up the one on the top ( which is the last one to be placed on top of the stack ). Documentation of the various operations and the stages a stack

Queues First in First Out data structure (FIFO). Like people waiting to buy tickets in a queue - the first one to stand in the queue, gets the ticket first and gets to leave the queue first. Documentation of the various operations

Single Linked List A self referential data structure. A list of elements, with a head and a tail; each element points to another of its own kind.

Double Linked ListA self referential data structure. A list of elements, with a head and a tail; each element points to another of its own kind in

passes through when elements are inserted or deleted. C program to help you get an idea of how a stack is implemented in code.

and the stages a queue passes through as elements are inserted or deleted. C Program source code to help you get an idea of how a queue is implemented in code.

front of it, as well as another of its own kind, which happens to be behind it in the sequence.

Circular Linked List Linked list with no head and tail - elements point to each other in a circular fashion.

1.

Tree Data Structures

Binary Search Trees A basic form of tree data structures. Inserting and deleting elements in them. Different kind of binary tree traversal algorithms.

Heaps - A tree like data structure where every element is lesser (or greater) than the one above it. Heap formation, sorting using heaps in O(n log n) time.

Height Balanced Trees - Ensuring that trees remain balanced to optimize complexity of operations which are performed on them.

Graphs and Graph Algorithms

Depth First Search - Traversing through a graph using Depth First Search in which unvisited neighbors of the current vertex are pushed into a stack and visited in that order.

Breadth First Search - Traversing through a graph using Breadth First Search in which unvisited neighbors of the current vertex are pushed into a queue and then visited in that order.

Minimum Spanning Trees: Kruskal AlgorithmFinding the Minimum Spanning Tree using the Kruskal Algorithm which is a greedy technique.

Minumum Spanning Trees: Prim's AlgorithmFinding the Minimum Spanning Tree using the Prim's Algorithm.

Introducing the concept of Union Find.

Dijkstra Algorithm for Shortest PathsPopular algorithm for finding shortest paths : Dijkstra Algorithm.

Floyd Warshall Algorithm for Shortest PathsAll the all shortest path algorithm: Floyd Warshall Algorithm

Bellman Ford Algorithm Another common shortest path algorithm : Bellman Ford Algorithm.

Popular Algorithms in Dynamic Programming

Dynamic Programming A technique used to solve optimization problems, based on identifying and solving sub-parts of a problem first.

Integer Knapsack problem An elementary problem, often used to introduce the concept of dynamic programming.

Matrix Chain MultiplicationGiven a long chain of matrices of various sizes, how do you parenthesize them for the purpose of multiplication - how do you chose which ones to start multiplying first?

Longest Common Subsequence Given two strings, find the longest common sub sequence between them.

Dynamic Programming Algorithms covered previously: Insertion Sort, Floyd Warshall AlgorithmAlgorithms which we already covered, which are example of dynamic programming.

Greedy Algorithms

Elementary cases : Data Compression

Fractional Knapsack Problem, Task Scheduling - Elementary problems in Greedy algorithms - Fractional Knapsack, Task Scheduling. Along with C Program source code.

using Huffman TreesCompression using Huffman Trees. A greedy technique for encoding information.

Commonly Asked Programming Interview Questions - from Microsoft/Google/Facebook/Amazon interviews Programming Interview Questions with Solutions - Microsoft, Google, Facebook, Amazon

A Collection of C Programs C Programs - Exploring various things which can be done in C

Miscellaneous C Programs1. 1 Computing the Area of a Circle in

C 2. 2 C Program to check for Armstrong

Numbers3. 3 C Program for Bezier Curves4. 4 C Program implementing the

Bisection Method ( Numerical Computing )

5. 5 C Program demonstrating the use of Bitwise Operators

6. 6 C Program for an Expression Evaluator

7. 7 C Program to demonstrate File Handling Functions

8. 8 C Program to demonstrate the Gaussian Elimination Method

9. 9 C Program to compute the GCD (HCF) of two numbers

10. 10 C Program to solve the Josephus Problem

11. 11 C Program to demonstrate operations on Matrices

12. 12 C Program implementing the Newton Raphson Method (Numerical Computing)

13. 13 C Program to check whether a string is a palindrome or not

14. 14 C Program to print the Pascal Triangle

15. 15 C Program to display Prime Numbers using the sieve of Eratosthenes

16. 16 C Program for the Producer - Consumer Problem

17. 17 C Program for the Reader - Writer Problem

18. 18 C Program to demonstrate the Dining Philosopher problem

19. 19 C Program to reverse the order of words in a sentence

20. 20 C Program to reverse a string21. 21 C Program to demonstrate the

values in the series expansion of exp(x),sin(x),cos(x),tan(x)

22. 22 C Program to demonstrate common operations on Sets

23. 23 C Program to solve Simultaneous Linear Equations in two variables

24. 24 C program to display the total number of words,the number of unique words and the frequency of each word

25. 25 C program to display the IP address

26. 26 C program implementing the Jacobi method (Numerical Computing)

Functional Programming Principles and Techniques Functional Programming - A General Overview

Using the Functional Programming paradigm with a regular programming language like Ruby

Databases - A Quick Introduction To SQL - Sample Queries demonstrating common commandsIntroduction to SQL- A few sample queries - A Case Study - Coming up with a Schema for Tables -Taking a look at how the schema for a database table is defined, how different fields require to be defined. Starting with a simple "case study" on which the following SQL tutorials will be based.

Introduction to SQL- A few sample queries : Creating Tables (CREATE) Creating tables, defining the type and size of the fields that go into it.

Introduction to SQL - A few sample queries : Making Select Queries Elementary database queries - using the select statement, adding conditions and clauses to it to retrieve information stored in a database.

Introduction to SQL - A few sample queries : Insert, Delete, Update, Drop, Truncate, Alter Operation Example of SQL commands which are commonly used to modify database tables.

Introduction to SQL - A few sample queries: Important operators - Like, Distinct, Inequality, Union, Null, Join, Top Other Important SQL operators.

Introduction to SQL- A few sample queries: Aggregate Functions - Sum, Max, Min, Avg - Aggregate functions to extract numerical features about the data.

Introduction To Networking

Client Server Program in Python

A basic introduction to networking and client server programming in Python. In this, you will see the code for an expression calculator . Clients can sent expressions to a server, the server will evaluate those expressions and send the output back to the client.

Introduction to Basic Digital Image Processing Filters

Introductory Digital Image Processing filters

Low-pass/Blurring filters, hi-pass filters and their behavior, edge detection filters in Matlab . You can take a look at how different filters transform images. Matlab scripts for these filters.

An Introduction to Graphics and Solid Modelling 3D Modelling in Solid Works - Part I

A Tutorial on 3D Modelling in SolidWorks - Part II

Autodesk 3DS-Max Autodesk 3DS Max - Part II

Intro to Google Sketchup

Quick Introduction to Open GL (with C++) - Part I

Quick Introduction to Open GL (with C++) - Part II

Electrical Science and Engineering

Introduction to DC Circuits

Circuit Theory 1a- Introduction to Electrical Engineering, DC Circuits, Resistance and Capacitance, Kirchoff Law Resistors, Capacitors, problems related to these.

Circuit Theory 1b - More solved problems related to DC Circuits with Resistance and CapacitanceCapacitors, computing capacitance, RC Circuits, time constant of decay, computing voltage and electrostatic energy across a capacitance

Circuit Theory 2a - Introducing InductorsInductors, inductance, computing self-inductance, flux-linkages, computing energy stored as a magnetic field in a coil, mutual inductance, dot convention, introduction to RL Circuits and decay of an inductor.

Circuit Theory 2b - Problems related to RL, LC, RLC circuits Introducing the concept of oscillations. Solving problems related to RL, LC and RLC circuits using calculus based techniques.

Circuit Theory 3a - Electrical Networks and Network Theorems Different kind of network elements: Active and passive, linear and non-linear, lumped and distributed. Voltage and current sources. Superposition

Circuit Theory 3b - More network theorems, solved problems More solved problems and examples related to electrical networks. Star and Delta network transformations, maximum power

theorem, Thevenin (or Helmholtz) theorem and problems based on these.

transfer theorem, Compensation theorem and Tellegen's Theorem and examples related to these.

Introduction to Digital Electronic Circuits and Boolean logic

Introduction to the Number System : Part 1 Introducing number systems. Representation of numbers in Decimal, Binary,Octal and Hexadecimal forms. Conversion from one form to the other.

Number System : Part 2 Binary addition, subtraction and multiplication. Booth's multiplication algorithm. Unsigned and signed numbers.

Introduction to Boolean Algebra : Part 1 Binary logic: True and false. Logical operators like OR, NOT, AND. Constructing truth tables. Basic postulates of Boolean Algebra. Logical addition, multiplication and complement rules. Principles of duality. Basic theorems of boolean algebra: idempotence, involution, complementary, commutative, associative, distributive and absorption laws.

Boolean Algebra : Part 2De-morgan's laws. Logic gates. 2 input and 3 input gates. XOR, XNOR gates. Universality of NAND and NOR gates. Realization of Boolean expressions using NAND and NOR. Replacing gates in a boolean circuit with NAND and NOR.

Understanding Karnaugh Maps : Part 1 Introducing Karnaugh Maps. Min-terms and Max-terms. Canonical expressions. Sum of products and product of sums forms. Shorthand notations. Expanding expressions in SOP and POS Forms ( Sum of products and Product of sums ). Minimizing

Karnaugh Maps : Part 2Map rolling. Overlapping and redundant groups. Examples of reducing expressions via K-Map techniques.

Introduction to Combinational Circuits : Part 1Combinational circuits: for which logic is entirely dependent of inputs and nothing else. Introduction to Multiplexers, De-multiplexers, encoders and decoders.Memories: RAM and ROM.

Combinational Circuits : Part 2 Static and Dynamic RAM, Memory organization.

boolean expressions via Algebraic methods or map based reduction techniques. Pair, quad and octet in the context of Karnaugh Maps.

Different kinds of ROM - Masked ROM, programmable ROM.

Introduction to Sequential Circuits : Part 1 Introduction to Sequential circuits. Different kinds of Flip Flops. RS, D, T, JK. Structure of flip flops. Switching example. Counters and Timers. Ripple and Synchronous Counters.

Sequential Circuits : Part 2ADC or DAC Converters and conversion processes. Flash Converters, ramp generators. Successive approximation and quantization errors.

Clockwise : Fractal Geometry in Nature , Projectile Motion , A graph , An array being sorted