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Linear Algebra

Calculus

Analytic Geometry

Ordinary Differential Equations

Dynamics & Statics

Vector Analysis

Algebra

Real Analysis

Complex Analysis

Linear Programming

Partial differential equations

Numerical Analysis and Computer programming

All Exams > UPSC > Mathematics Optional Notes for UPSC > All Videos

All videos of Mathematics Optional Notes for UPSC for UPSC CSE Exam

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

Rank of Matrix

17:17 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

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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

Real Numbers

24:46 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

Reimann's Sum

08:56 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

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

03:40 min

Equation of Sphere

09:29 min

General Equation of Cone

07:58 min

Hyperboloid of One and Two Sheets

05:49 min

The Ellipsoid

05:55 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

Cyclic Group

12:27 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

Factor Groups

16:59 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

Ring Examples

07:18 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

Inegral Domain

07:34 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

21:41 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

Power Series

31:55 min

Taylor Series

31:28 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

Laurent Series

43:51 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

PDE: Surfaces

19:27 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

15:02 min

Trapezoidal Rule

12:10 min

Trapezoidal Rule for Integration

08:07 min

Simpson's Rule

12:11 min

Gaussian Quadrature Formula

08:51 min

Euler Method

07:03 min

Euler's Method with Examples

20:50 min

Runge Kutta Method

12:29 min

Binary Numbers

11:21 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

Mathematics Optional Notes for UPSC

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Video lectures for Mathematics Optional Notes for UPSC 2025 is part of UPSC CSE exam preparation. The videos have been prepared according to the UPSC CSE exam syllabus. The Video lectures, notes, tests & MCQs are made for UPSC CSE 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.

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