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 !
Our Index of Tutorials for all the topics
| Introduction to Complex Numbers|
Introduction to Complex Numbers and iota. Arg-and 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.
Series and Progressions
Arithmetic, 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.
The Principle of Mathematical Induction
Introductory problems related to Mathematical Induction.
Introducing 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 for these. Graphs and plots of quadratic equations.
Quadratic inequalities. Using factorization and visualization based methods.
| || Linear Algebra - Matrices Part I - A Tutorial with Examples 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 Algerba - Matrices Part II - A Tutorial with Examples, Problems and Solutions 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 - A Tutorial with Examples, Problems and Solutions 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 Equations in Multiple Variables - A Tutorial with Examples and Problems 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 - A Tutorial with Examples and Solved ProblemsTheory 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 SpacesProblems 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 about 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 - Linear Transformations, Operators and Maps Solved examples and problems related to linear transformation, linear maps and operators and other concepts discussed theoretically in the previous tutorial.
Linear Algebra - Eigenvalues, Eigenvectors and Cayley Hamilton Theorem Eigenvalues, eigenvectors, Cayley Hamilton Theorem
Linear Algebra - Problems Based on Simultaneous Equations, Eigenvalues, EigenvectorsDemonstrating 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 Recurrence Relations Solving a recurrence relation, some more of system of equations.
| Introduction to Vectors - Zero Vectors, Unit Vectors, Coinitial , Collinear, Equal Vectors, Addition and Subtraction of Vectors, Scalar and Vector Multiplication 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.|
Vectors: Introductory Problems and Examples - Related to products, properties of vectors, proving geometric properties using vectors. Solved examples and problem sets based on the above concepts.
Applying Vectors to Geometric Problems - Parametric Vectorial equation of a line and Plane, Condition for collinearity of three points, Shortest distance between two lines, Perpendicular distance of a point from a plane or line, Angles between lines and planes 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.
Vector Applications in 2D and 3D Geometry: Solved Problems and Examples - Shortest and Perpendicular Distances, Proving properties of Triangles, Tetrahedrons and Parallelograms using Vector methods Solved examples and problem sets based on the above concepts.
Vector Differential And Integral Calculus: Theory and Definitions - Differentiation of Vectors, Introduction to Div, Curl, Grad; Vector Integral Calculus; Green’s theorem in the plane; Divergence theorem of Gauss, etc.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.
Vector Differential And Integral Calculus: Solved Problem Sets - Differentiation of Vectors, Div, Curl, Grad; Green’s theorem; Divergence theorem of Gauss, etc. Solved examples and problem sets based on the above concepts.
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 - 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.
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).
Functions, Limits and Continuity - Solved Problem Set III - Continuity and Intermediate Value Theorems - Problems related to Continuity, intermediate value theorem.
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.
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 .
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.
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.
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.
Differential Calculus - Solved Problems Set V - Curve Sketching, Parametric Curves - More examples of investigating and sketching curves, parametric representation of curves.
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 - 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 - 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 - Solved Problems Set III - Reduction Formulas, Using Partial FractionsI- Examples and solved problems - Reduction formulas, reducing the integrand to partial fractions, more of definite 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 - 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.
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 - 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 - Solved Problems - Set III - More Differential Equations - More complex cases of 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
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.
Electrostatics and Electromagnetism
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 - 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.
Insertion Sort - Another quadratic time sorting algorithm - an 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.
Selection Sort - Another quadratic time sorting 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.
Shell Sort- An inefficient but interesting 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 Sort- Efficient sorting algorithm which runs in O(n log n) time. Uses the Heap data structure.
Binary Search Algorithm- Commonly 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 passes through when elements are inserted or deleted. C program to help you get an idea of how a stack is implemented in code.
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 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.
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 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 in 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.
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
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 problemAn elementary problem, often used to introduce the concept of dynamic programming.
Matrix Chain Multiplication Given 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 Algorithm Algorithms which we already covered, which are example of dynamic programming.
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
Functional Programming Principles and Techniques
Introduction to Ruby
Introduction to Ruby and some playing around with the Interactive Ruby Shell (irb)
Introduction to Ruby - Conditional statements and Modifiers: If-then, Unless, Case
Introduction to Ruby Comments - Single and Multi-Line comments
Introduction to Ruby Loops - Using While, Until, For, Break, Next , Redo, Retry
Introduction to Ruby - Arrays - Sorting, Filtering (Select), Transforming, Multi-Dimensional Arrays
Introduction to Ruby - Strings
Introduction to Ruby - Making a Script Executable
Introduction to Ruby - Regular Expressions, Match, Scan
Introduction to Ruby - Computing Factorials Recursively : An Example of Recursion
Introduction to Ruby - Binomial Coefficients (nCr) : An Example of Recursion
Introduction to Ruby - Computing a Power Set : An Example of Recursion
Introduction to Ruby - Towers of Hanoi : An Example of Recursion
Introduction to Ruby - Strings: Substitution, Encoding, Built-In Methods
Basic Data Structures & Collections With Ruby
Databases - A Quick Introduction To SQL - Sample Queries demonstrating common commands
Introduction 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
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 Capacitance- Capacitors, computing capacitance, RC Circuits, time constant of decay, computing voltage and electrostatic energy across a capacitance
Circuit Theory 2a - Introducing Inductors- Inductors, 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 theorem, Thevenin (or Helmholtz) theorem and problems based on these.
Circuit Theory 3b - More network theorems, solved problems- More solved problems and examples related to electrical networks. Star and Delta network transformations, maximum power 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 1Binary 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 boolean expressions via Algebraic methods or map based reduction techniques. Pair, quad and octet in the context of Karnaugh Maps.
Karnaugh Maps : Part 2 Map rolling. Overlapping and redundant groups. Examples of reducing expressions via K-Map techniques.
Introduction to Combinational Circuits : Part 1 Combinational circuits: for which logic is entirely dependent of inputs and nothing else. Introduction to Multiplexers, De-multiplexers, encoders and decoders.Memories: RAM and ROM. 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 2
ADC or DAC Converters and conversion processes. Flash Converters, ramp generators. Successive approximation and quantization errors.
Test Preparations - the Olympiads, Board Exams, and the IIT JEE
Infographic Insights about India
Visualizing Energy Statistics for India
Clockwise : Fractal Geometry in Nature , Projectile Motion , A graph , An array being sorted