Video Lectures of Mathematics Optional Notes for UPSC for UPSC CSE Exam
Watch free video lectures for Mathematics Optional Notes for UPSC covering all chapters/topics as per the latest syllabus in 2026. These concept videos, recorded lectures, and audio notes are designed to help UPSC CSE students understand every topic clearly — anytime, anywhere. Start learning on EduRev.
Linear Algebra
| What is a Vector Space? 06:58 min |  |
| Vector Spaces over R and C 08:01 min | |
| Introduction to Linerar Space 15:46 min | |
| Linear Dependence and Independence 08:16 min | |
| Linear Independence 12:56 min | |
| Subspaces- Linear Algebra 13:04 min | |
| Subspace Proof Example 05:42 min | |
| Linear Independence and Bases 08:40 min | |
| Linear Algebra - Dimensions 11:14 min | |
| Basis and Dimensions 10:06 min | |
| Linear Transformation and Matrices 10:59 min | |
| Inverse matrices, column space and null space 12:09 min | |
| Rank and Nullity of a Matrix ( Row Echeleon ) 22:10 min | |
| Linear Transformation on Vector Spaces 09:11 min | |
| Matrix Transormations 11:32 min | |
| Linear Transformations 09:19 min | |
| Introduction to Matrices 11:23 min | |
| Adding and Subtracting Matrices 06:12 min | |
| Scalar Multiplication of Matrices and Matrix Operations 06:37 min | |
| Multiplying Matrices 17:40 min | |
| Row Echelon Form of the Matrix 11:11 min | |
| Reduced Row Echelon Form of Matrix 08:44 min | |
| Gauss Jordan Elimination & Reduced Row Echelon Form 10:51 min | |
| Similarity of Matrices 19:39 min | |
| Finding Rank of a Matrix 03:25 min | |
| Inverse of 3X3 Matrix 15:21 min | |
| Inverse of a Matrix 04:27 min | |
| Inverse Matrices and their Properties 12:00 min | |
| Inverse of Matrix: Example 06:22 min | |
| System of Linear Equations- 1 04:04 min | |
| System of Linear Equations- 2 04:16 min | |
| Solution Sets for System of Equations 11:25 min | |
| Eigenvalues and Eigenvectors 17:16 min | |
| Finding of Eigenvalues and Eigenvectors 17:10 min | |
| Introduction to Characteristic Polynomial, Eigenvalues and Eigenvectors 17:53 min | |
| Characteristic Polynomial 12:01 min | |
| Finding Characteristic Polynomial of a Matrix 03:54 min | |
| The Cayley Hamilton Theorem 21:38 min | |
| Cayley Hamilton Theorem 31:23 min | |
| Overview of Symmetric Matrix 07:00 min | |
| Overview of Skew Symmetric Matrix 06:51 min | |
| Symmetric Matrix 25:51 min | |
| Symmetric and Skew Symmetric Matrix 11:31 min | |
| What is Hermitian Matrix 06:00 min | |
| What is Skew Hermitian Matrix 02:28 min | |
| Theorem on Hermitian and Skew Hermitian Matrix 09:00 min | |
| Hermitian Matrix 46:55 min | |
| Hermitian, Normal and Unitary Matrix 05:52 min | |
| Orthogonal Matrix 04:44 min | |
| Proving a Matrix is Orthogonal 03:19 min | |
| Problem on Unitary Matrix 07:10 min | |
| Problem on Orthogonal Matrix 05:22 min | |
| Null space and Nullity of a Matrix 07:53 min | |
| Eigenvalues and EigenVectors 17:10 min | |
Calculus
| Fun Video: Overview of Limits and Derivatives 10:02 min | |
| Examples of Limits of Polynomials (with Explanation) 05:42 min | |
| Limits: Examples Part 1 10:03 min | |
| Introduction to Limits: Part 1 11:32 min | |
| Limits By Using Standard Formula 38:07 min | |
| Limits of Non-Algebraic Functions 41:32 min | |
| Advanced Exponential Limits - 1 38:31 min | |
| Advanced Exponential Limits - 2 34:07 min | |
| Newton-Leibniz Formula for Limits 29:17 min | |
| Limits and Continuity Definitions 12:06 min | |
| Limits and Continuity Graphical Interpretation of Continuity 11:20 min | |
| Limits and Continuity Difference between Not Defined, infinity, 0 and approaching 0 08:21 min | |
| Limits and Continuity Indeterminate Forms 03:45 min | |
| Limits and Continuity Important Formulas 05:19 min | |
| Limits and Continuity Examples of Direct Substitution, Factorization, Rationalization 19:18 min | |
| Limits and Continuity Examples of sinx/x, tanx/x, (e^x-1)/x, (a^x-1)/x problems 13:33 min | |
| Limits and Continuity Examples of log(1+x)/x, infinity/infinity problems 09:43 min | |
| Limits and Continuity Series Expansion 14:05 min | |
| Limits and Continuity L'Hospital Rule and other topics 07:45 min | |
| Video: Continuity of a Function 08:28 min | |
| Continuity & Differentiability 27:32 min | |
| Proof of theorem stating Differentiability implies Continuity 11:40 min | |
| Problems on Continuity & Differentiability - 1 49:29 min | |
| Problems on Continuity & Differentiability - 2 54:30 min | |
| Short Tricks for Continuity & Differentiability 54:29 min | |
| Continuity & Differentiability- 2 28:38 min | |
| Continuity & Differentiability- 3 20:51 min | |
| Mean Value Theorem 06:37 min | |
| Maxima and Minima 04:30 min | |
| Examples : Absolute maxima and minima 13:15 min | |
| Finding maxima and Minima Using Second Derivative Test (with Example) 08:15 min | |
| Introduction to Orthogonal Matrix 03:08 min | |
| Functions of Real Variables 13:34 min | |
| Introduction to Limits 20:20 min | |
| Limits and L'Hospital's Rule 18:27 min | |
| Limits and Limit Laws in Calculus 12:49 min | |
| Continuity and Differentiability 32:48 min | |
| Understanding Mean Value Theorem 19:40 min | |
| Taylor's Theorem 07:01 min | |
| Taylor's Remainder Theorem 14:08 min | |
| Indeterminate Forms 07:31 min | |
| Horizontal and Vertical Asymptotes 31:36 min | |
| Curve Sketching: First and Second Derivative 41:29 min | |
| Functions of Several Variables 20:14 min | |
| Introduction to Partial Derivatives 10:56 min | |
| Partial Derivatives 60:33 min | |
| Langrage's Multipliers 33:46 min | |
| What is Jacobian 27:14 min | |
| How to Find Jacobian Matrix 09:58 min | |
| Finding Definite Integral using Reimann Sums 13:18 min | |
| Indefinite Integral 29:00 min | |
| Improper Integrals 13:56 min | |
| Double and Triple Integrals 15:29 min | |
| Double Integrals-1 10:29 min | |
| Double Integrals-2 09:51 min | |
| Double Integrals-3 08:04 min | |
| Triple Integrals-1 10:38 min | |
| Triple Integrals-2 07:26 min | |
| Triple Integrals-3 11:48 min | |
| Mean Value Theorem 19:40 min | |
| Taylor’s series 09:34 min | |
| Maxima and minima(in one variable) 06:17 min | |
| Limit(Two or three variable) 19:04 min | |
| Partial Derivatives 09:59 min | |
| Double and Triple Integrals 15:29 min | |
Analytic Geometry
| Different Forms of Line 19:33 min | |
| Two-Point Form of Line 19:16 min | |
| Straight Line in Perpendicular Form 29:04 min | |
| Different Forms of Straight Line 20:44 min | |
| Problems on various Forms of Straight Line 18:55 min | |
| Reduction of Different Forms of Straight Line 18:33 min | |
| Parallel & Perpendicular Lines 16:06 min | |
| Distance of Point From Straight Line 17:48 min | |
| Family of Lines 16:17 min | |
| Problems on Family of Lines 18:59 min | |
| Concurrency of Three Lines 13:58 min | |
| Mirror Image of Point 24:49 min | |
| Equation of Reflected Ray 14:50 min | |
| Problems on Reflection of Ray 13:26 min | |
| Coordinate System in a Plane- 1 46:55 min | |
| Coordinate System in a Plane- 2 35:54 min | |
| Coordinate System in a Plane- 3 41:27 min | |
| Coordinate System in a Plane- 4 45:02 min | |
| Surface Area of Revolution By Integration 30:36 min | |
| Introduction to Polar Coordinates 22:29 min | |
| Point Plotting in 3D Coordinate System 07:27 min | |
| Second Degree Equation with 3 Variables 06:09 min | |
| Quadratic Equation with 3 Variables 10:53 min | |
| Understanding Equation of Sphere 11:48 min | |
| Equation of Sphere 09:29 min | |
| General Equation of Cone 07:58 min | |
| Hyperboloid of One and Two Sheets 05:49 min | |
| Hyperboloid of One Sheets 06:31 min | |
| Hyperboloid of Two Sheets 08:37 min | |
| The Hyperbolic Paraboloid 07:46 min | |
| The Elliptical Paraboloid 06:02 min | |
| Equation of Sphere 09:41 min | |
| The Elliptical Cone 07:20 min | |
Ordinary Differential Equations
| Introduction to Differential Equations (includes Order and Degree of a Differential Equation) 09:38 min | |
| Differential Equations Representing a Family of Curves 07:11 min | |
| Homogeneous Differential Equations and their Solution 07:04 min | |
| Solving Differential Equations by Variables Separable Method 09:06 min | |
| General solution and Particular Solution of a Differential Equation 04:54 min | |
| Linear Differential Equations and their Solution using Integrating Factor 05:37 min | |
| Formation of a Differential Equation whose General Solution is Given 04:48 min | |
| Formulation and Interpretation of Differential Equations 09:04 min | |
| First Order Differential Equation 07:49 min | |
| First Order Linear Differential Equation 22:28 min | |
| Method of Integrating Factors 05:07 min | |
| Integrating Factor 09:14 min | |
| Orthogonal Trajectory in Differential Equations 11:25 min | |
| Clairaut's Equation 08:16 min | |
| How to Solve Clairaut's Equation 09:43 min | |
| Linear Second Order Homogeneous Differential Equations with Constant Coef 10:16 min | |
| Solved Example: Linear Second Order Differential Equation 07:23 min | |
| Initial Value Problem 07:13 min | |
| Introduction to Complimentary Function and Particular Integral 14:54 min | |
| Solved Example: Complimentary Function and Particular Integral 16:40 min | |
| Second Order Linear Differential Equation 25:17 min | |
| Second-order linear equations with variable coefficients 56:23 min | |
| Cauchy Euler Equation 14:05 min | |
| Variation of Parameters 11:36 min | |
| Variation of Parameters (Non Homogenous) 09:58 min | |
| Laplace Transforms 19:27 min | |
| Inverse Laplace Transform Example using Partial Fractions 08:53 min | |
| Inverse Laplace Transform 12:44 min | |
| Laplace Transform of Elementary Functions 41:48 min | |
| Initial Value Problem: Second Order Differential Equation 26:17 min | |
Dynamics & Statics
| Simple Harmonic Motion 26:23 min | |
| Compound Pendulum 08:11 min | |
| Equivalent Length of a Simple Pendulum 11:31 min | |
| Energy of Oscillation in Angular SHM 05:05 min | |
| Cases of Spring Combinations in SHM 07:21 min | |
| SHM of Free Bodies in Absence of External Forces 06:42 min | |
| Motion in a Plane 42:39 min | |
| Projectile Motion 73:15 min | |
| Short Tricks: Projectile Motion 02:13 min | |
| Velocity of a Particle During Projectile 03:05 min | |
| Position and Trajectory of Projectile 05:10 min | |
| Projectile in Moving Reference Frame 07:04 min | |
| Projectile on Inclined Plane 11:36 min | |
| Work and Energy 11:23 min | |
| Principle of Work & Energy 15:08 min | |
| Work-Energy Theorem 04:33 min | |
| Potential Energy 12:37 min | |
| The Law of Conservation of Energy 10:10 min | |
| Kepler's Laws of Planetary Motion 01:47 min | |
| Kepler's Law & Proofs 36:41 min | |
| Kepler's Laws of Planetary Motion-Second Law 09:12 min | |
| Kepler's Laws of Planetary Motion-Third Law 04:49 min | |
| Stable, Unstable, and Neutral Equilibrium 01:05 min | |
| Introduction to Rectilinear Motion 07:29 min | |
| Rectilinear Motion Problems 16:14 min | |
| Constrained Motion-1 09:11 min | |
| Constrained Motion-2 02:01 min | |
| Static and Kinetic Friction 24:49 min | |
| Static Friction 04:14 min | |
| Principle of Virtual Work 12:08 min | |
| Stable and Unstable Equilibrium Points 07:22 min | |
| Equilibrium of Rigid Bodies 3D force Systems 10:14 min | |
Vector Analysis
| Scalor and Vector Fields 08:53 min | |
| Scalor and Vector Fields 13:32 min | |
| Higher Order Derivatives 10:51 min | |
| Vector Identities 09:59 min | |
| Finding The Vector Equation of a Line and Symmetric & Parametric Equations 11:37 min | |
| Finding The Equation of a Plane Given a Point and Perpendicular Normal Vector 07:37 min | |
| Serret-Frenet's formulae 24:39 min | |
| Gauss's theorem 26:53 min | |
| Stokes' theorem 23:54 min | |
| Green's First Identity 05:25 min | |
| Green's Second Identity 03:12 min | |
| Torsion of a Space Curve 11:00 min | |
| Curvature of a Space Curve 11:47 min | |
| Vector Functions and Space Curves 22:36 min | |
| Vector Fields, Divergence, and Curl 15:36 min | |
| Partial Derivatives of Vector Fields 08:34 min | |
| Derivation of Gradient in Cylindrical coordinates 08:59 min | |
| Polar Coordinates in Divergence and Curl 16:23 min | |
Algebra
| Group Definition 11:15 min | |
| Understanding Subgroups 15:51 min | |
| Cosets and their Properties 15:21 min | |
| Lagrange's Theorem 14:05 min | |
| Normal and Quotient Subgroups 11:24 min | |
| Normal Subgroups 09:14 min | |
| Group Homomorphism 10:04 min | |
| The Kernel of a Group Homomorphism 04:53 min | |
| Homomorphism and Isomorphism Theorems 12:47 min | |
| Natural Proof of First Isomorphism Theorem 13:08 min | |
| Permutation Group 15:31 min | |
| Properties of Permutation 16:00 min | |
| Cayley's Theorem 04:40 min | |
| Definiton of Ring 06:51 min | |
| Units in a Ring 07:14 min | |
| Intution Behind Subrings 10:22 min | |
| Subring Proof Examples 12:21 min | |
| Ideals in Ring Theory 11:57 min | |
| Ideals and Factor Rings 19:58 min | |
| Examples of Ideals and Factor Rings 13:34 min | |
| Ring Homomorphism 10:01 min | |
| Principle Ideal Domain 28:05 min | |
| Euclidean Domains 26:15 min | |
| Unique Factorization Domain-1 34:55 min | |
| Unique Factorization Domain-2 34:18 min | |
| Field Definition 08:06 min | |
| Quotient Fields 27:27 min | |
Real Analysis
| Ordered Fields and the Real Number System 10:51 min | |
| Least Upper Bound Property 10:36 min | |
| Definition of the Limit of a Sequence 13:59 min | |
| Limits of the Sequence 11:20 min | |
| Introduction to Cauchy Sequence 15:53 min | |
| Sequence is Cauchy if and only if it Converges 24:23 min | |
| Completeness of the Real Number 15:29 min | |
| The Real Number Line (completeness, cardinality, and measure) 07:42 min | |
| Convergence and Divergence - Introduction to Series 16:18 min | |
| Series Convergence Tests 15:36 min | |
| Absolute Convergence, Conditional Convergence, and Divergence 13:07 min | |
| Determining Absolute or Conditional Convergence 16:04 min | |
| Checking for Absolute Convergence 19:15 min | |
| Rearrange a Series 14:20 min | |
| Rearrangements of absolutely convergent series 11:19 min | |
| Continuity and Examples 05:33 min | |
| What is Uniform Continuity 06:23 min | |
| Uniform Continuity: Basics 07:14 min | |
| Uniform Continuity: Example 09:39 min | |
| Continous Function On Compact Sets 13:17 min | |
| Introduction to the Riemann Integral 38:16 min | |
| Real Analysis - Riemann Integrability 19:18 min | |
| Improper Integrals- 1 18:34 min | |
| Improper Integrals- 2 10:00 min | |
| Fundamental theorems of integral calculus 20:46 min | |
| The Fundamental Theorem of Calculus 09:38 min | |
| Uniform convergence of sequences and series of functions 50:47 min | |
| Pointwise and Uniform Convergence Visualized 07:51 min | |
| Proof: Differentiability Theorem on Uniform convergence of Sequence of Functions 24:20 min | |
| Partial Derivatives of Functions of Two And Three Variables 09:29 min | |
| Local Maxima and Minima of Function 14:18 min | |
Complex Analysis
| Conformal Mapping in Complex Variables 14:19 min | |
| Application of Conformal Mapping to Potential Theory 44:07 min | |
| Conformal mapping from disk to disk and angular region to disk 39:46 min | |
| Conformal mapping from half plane to disk and half plane to half plane-I 44:53 min | |
| Analytic Function 42:00 min | |
| Cauchy-Riemann Equations 50:14 min | |
| Harmonic Functions, Harmonic Conjugates and Milne's Method 35:07 min | |
| Cauchy's Theorem - I 55:50 min | |
| Cauchy's Theorem-II 30:32 min | |
| Complex Integration 42:41 min | |
| Cauchy's Integral Formula for the Derivatives of Analytic Function 58:30 min | |
| Sequences and Series 32:49 min | |
| Uniform Convergence of Series 30:10 min | |
| Zeros and Singularities of an Analytic Function 42:16 min | |
| Residue at a Singularity 35:24 min | |
| Meromorphic Functions 42:53 min | |
| Residue Theorem 49:09 min | |
| Evaluation of real integrals using residues-I 39:20 min | |
| Evaluation of real integrals using residues-II 56:08 min | |
| Evaluation of real integrals using residues-III 27:32 min | |
| Evaluation of real integrals using residues-IV 36:38 min | |
| Evaluation of real integrals using residues-V 36:04 min | |
| Complex Analysis: Contour integration 20:17 min | |
Linear Programming
| Linear Programming 33:20 min | |
| Solve a Linear Programming Problem Using the Graphical Method 11:49 min | |
| Solution of LPP using Simplex Method 27:14 min | |
| Solving Linear Programming Using Dual Simplex Method 11:07 min | |
| Transportation Problem 06:41 min | |
| Solving Transportation Problems with Linear Programming 09:31 min | |
| Assignment Problem - Linear Programming LP Model and Excel Model 12:28 min | |
Partial differential equations
| Cauchy Problem for First Order PDEs 30:50 min | |
| First order equations in two variables-1 36:04 min | |
| Method of Separation of Variables for Heat Equation- 1 11:09 min | |
| Method of Separation of Variables for Heat Equation- 2 10:51 min | |
| First Order Partial Differential Equation: Solution of Lagrange Form 16:29 min | |
| Classification of Second Order PDEs 13:47 min | |
| First order equations in two variables-2 29:59 min | |
| First order equations in two variables-3 35:04 min | |
| First order equations in more than two variables-6 35:44 min | |
| First order equations in more than two variables-7 42:43 min | |
| Classification - 1 34:13 min | |
| Laplace and Poisson equations-1 32:12 min | |
| Laplace and Poisson equations-2 31:31 min | |
| Laplace and Poisson equations-3 24:09 min | |
| One dimensional heat equation-1 32:07 min | |
| One dimensional heat equation-2 27:03 min | |
| How to Solve Quasi-Linear PDEs 11:35 min | |
| Method of characteristics and PDE 20:06 min | |
| Cauchy's Method of Characteristics 22:51 min | |
| Canonical Form: Second Order PDE 43:46 min | |
| Wave Equations: Vibrating String 57:54 min | |
Numerical Analysis and Computer programming
| Gauss Elimination Method 17:18 min | |
| Gauss-Seidel Method 15:48 min | |
| Numerical Solutions of ODEs using Picard Method 11:03 min | |
| Numerical solutions of ODEs using modified Euler method 13:24 min | |
| Numerical Solutions of ODEs using Runge-Kutta Method 14:20 min | |
| Introduction to Numerical Analysis 50:23 min | |
| Interpolating Polynomials 49:37 min | |
| Polynomial Approximation 48:35 min | |
| Error in the Interpolating polynomial 49:45 min | |
| Properties of Divided Difference 50:01 min | |
| Cubic Spline Interpolation 47:47 min | |
| Cubic Hermite Interpolation 50:16 min | |
| Piecewise Polynomial Approximation 47:58 min | |
| Numerical Integration: Basic Rules 49:41 min | |
| Composite Numerical Integration 46:42 min | |
| Gauss 2-point Rule: Error 45:21 min | |
| Gauss 2-point Rule: Construction 54:50 min | |
| Numerical Differentiation 49:10 min | |
| Convergence of Gaussian Integration 48:49 min | |
| Gauss Elimination 46:38 min | |
| L U decomposition 46:26 min | |
| Gauss Elimination with partial pivoting 44:59 min | |
| Newton's Backward Interpolation 28:53 min | |
| Method of False Position: Reluga Falsi Method 07:36 min | |
| Newton's Method 10:41 min | |
| Newton Raphson Method 10:32 min | |
| Bisection Method 09:20 min | |
| Interpolation and Approximation 27:19 min | |
| Langrange's Formula 11:25 min | |
| Trapezoidal Rule 12:10 min | |
| Trapezoidal Rule for Integration 08:07 min | |
| Gaussian Quadrature Formula 08:51 min | |
| Euler's Method with Examples 20:50 min | |
| Runge Kutta Method 12:29 min | |
| Number System- Introduction 10:57 min | |
| Logic Gates, Truth Tables, Boolean Algebra AND, OR, NOT, NAND & NOR 54:07 min | |
| Unsinged Integers 06:04 min | |
| Signed Integers 06:25 min | |
