BMATS201

BMATS201 - MATHEMATICS-II FOR CSE STREAM

MODULE 1: INTEGRAL CALCULUS (8 Hours)

Topics:

1. Multiple Integrals

   • Double integrals
   • Triple integrals
   • Evaluation of double integrals
   • Evaluation of triple integrals

2. Change of Order of Integration

   • Converting limits
   • Changing order in double integrals
   • Problems

3. Changing to Polar Coordinates

   • Conversion from Cartesian to Polar
   • Evaluation using polar coordinates
   • Problems

4. Applications of Double Integrals

   • Finding Area using double integrals
   • Finding Volume using double integrals
   • Problems

5. Beta and Gamma Functions

   • Gamma Function γ(n):
     • Definition
     • Properties
     • γ(n+1) = n·γ(n)
     • γ(1/2) = √π
   • Beta Function β(m,n):
     • Definition
     • Properties
     • Relation: β(m,n) = γ(m)·γ(n)/γ(m+n)
   • Problems

Self-Study:

• Center of gravity
• Duplication formula

Applications:

• Antenna and wave propagation
• Calculation of optimum value in various geometries
• Analysis of probabilistic models

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MODULE 2: VECTOR CALCULUS (8 Hours)

Topics:

1. Scalar and Vector Fields

   • Definition
   • Examples
   • Difference

2. Gradient (∇φ or grad φ)

   • Definition
   • Properties
   • Physical interpretation
   • Problems

3. Directional Derivative

   • Definition
   • Formula
   • Maximum directional derivative
   • Problems

4. Curl (∇ × F or curl F)

   • Definition
   • Properties
   • Physical interpretation
   • Problems

5. Divergence (∇ · F or div F)

   • Definition
   • Properties
   • Physical interpretation
   • Problems

6. Solenoidal Vector Fields

   • Definition: div F = 0
   • Properties

7. Irrotational Vector Fields

   • Definition: curl F = 0
   • Properties

8. Vector Integration

   • Line integrals
   • Surface integrals
   • Volume integrals

9. Vector Integral Theorems

   • Green's Theorem (in plane)
   • Gauss Divergence Theorem
   • Stokes' Theorem
   • Verification problems

Self-Study:

• Conservative vector fields

Applications:

• Fluid flow analysis
• Electromagnetic field theory
• Computer graphics

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MODULE 3: LINEAR ALGEBRA (8 Hours)

Topics:

1. Vector Spaces

   • Definition
   • Examples
   • Subspaces

2. Linear Independence and Dependence

   • Definition
   • Testing for independence

3. Basis and Dimension

   • Basis vectors
   • Dimension of vector space
   • Standard basis

4. Inner Product Spaces

   • Inner product
   • Norm
   • Orthogonality

5. Gram-Schmidt Orthogonalization Process

   • Method
   • Problems

6. Linear Transformations

   • Definition
   • Matrix representation
   • Properties

7. Kernel and Range

   • Null space (Kernel)
   • Range space
   • Rank-Nullity theorem

8. Eigenvalues and Eigenvectors

   • Characteristic equation
   • Finding eigenvalues
   • Finding eigenvectors
   • Properties

9. Diagonalization

   • Conditions for diagonalization
   • Process
   • Problems

Self-Study:

• Cayley-Hamilton theorem

Applications:

• Machine learning algorithms
• Data compression
• Network analysis
• Google PageRank algorithm

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MODULE 4: NUMERICAL METHODS - I (8 Hours)

Topics:

1. Solution of Transcendental Equations

   Bisection Method:

   • Algorithm
   • Convergence
   • Problems

   Newton-Raphson Method:

   • Formula: x_{n+1} = x_n - f(x_n)/f'(x_n)
   • Convergence (quadratic)
   • Problems

   Regula-Falsi Method (False Position):

   • Algorithm
   • Convergence
   • Problems

2. Solution of System of Linear Equations

   Iterative Methods:

   Gauss-Jacobi Method:

   • Formula
   • Convergence criteria
   • Problems

   Gauss-Seidel Method:

   • Formula
   • Faster convergence than Jacobi
   • Problems

3. Interpolation

   Newton's Forward Difference Formula:

   • For equally spaced data
   • Forward difference table
   • Problems

   Newton's Backward Difference Formula:

   • For equally spaced data
   • Backward difference table
   • Problems

   Lagrange's Interpolation Formula:

   • For unequally spaced data
   • Formula
   • Problems

Self-Study:

• Fixed point iteration method

Applications:

• Root finding in computational problems
• Solving engineering equations
• Data interpolation in graphics

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MODULE 5: NUMERICAL METHODS - II (8 Hours)

Topics:

1. Numerical Differentiation

   • Forward difference formula
   • Backward difference formula
   • Central difference formula
   • Problems

2. Numerical Integration

   Trapezoidal Rule:

   • Formula
   • Error estimation
   • Problems

   Simpson's 1/3 Rule:

   • Formula
   • Error estimation
   • Problems

   Simpson's 3/8 Rule:

   • Formula
   • Problems

3. Ordinary Differential Equations (ODE)

   First Order ODEs:

   Euler's Method:

   • Formula: y_{n+1} = y_n + h·f(x_n, y_n)
   • Problems

   Modified Euler's Method:

   • Predictor-Corrector
   • Problems

   Runge-Kutta Method (4th Order):

   • RK4 formula
   • Most accurate
   • Problems

Self-Study:

• Taylor series method
• Milne's predictor-corrector method

Applications:

• Solving differential equations in physics
• Circuit analysis
• Population dynamics
• Heat transfer problems

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LABORATORY EXPERIMENTS (10-12 Sessions)

1. Multiple integration problems
2. Beta and Gamma function computations
3. Vector calculus operations (Gradient, Curl, Divergence)
4. Verification of vector theorems
5. Linear transformations and matrix operations
6. Eigenvalue and eigenvector computation
7. Bisection and Newton-Raphson methods
8. Gauss-Jacobi and Gauss-Seidel methods
9. Interpolation using Newton's and Lagrange's formulas
10. Numerical integration (Trapezoidal and Simpson's)
11. ODE solution using Euler's and RK methods
12. Gram-Schmidt orthogonalization

Software: MATLAB/Python/Scilab/Mathematica

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KEY FORMULAS:

Module 1:

• γ(n+1) = n!
• γ(1/2) = √π
• β(m,n) = γ(m)·γ(n)/γ(m+n)
• Polar conversion: x = r cos θ, y = r sin θ

Module 2:

• Gradient: ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k
• Divergence: ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z
• Curl: ∇×F determinant form

Module 3:

• Rank-Nullity: rank(T) + nullity(T) = n
• Eigenvalue equation: Av = λv
• Characteristic equation: |A - λI| = 0

Module 4:

• Newton-Raphson: x_{n+1} = x_n - f(x_n)/f'(x_n)
• Gauss-Seidel convergence faster than Jacobi

Module 5:

• Trapezoidal: ∫f(x)dx ≈ (h/2)[y_0 + 2(y_1+...+y_{n-1}) + y_n]
• Simpson's 1/3: ∫f(x)dx ≈ (h/3)[y_0 + 4(y_1+y_3+...) + 2(y_2+y_4+...) + y_n]
• Euler: y_{n+1} = y_n + hf(x_n,y_n)
• RK4: Most accurate numerical method

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EXAM PATTERN:

CIE: 50 marks (30 theory + 20 lab)
SEE: 50 marks
Duration: 3 hours
Questions: 10 (2 per module)
Answer: 5 questions (1 from each module)

Formula Handbook: Permitted in exam

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TEXTBOOKS:

1. B.S. Grewal - Higher Engineering Mathematics (44th Ed.)
2. E. Kreyszig - Advanced Engineering Mathematics (10th Ed.)
3. N.P. Bali - Engineering Mathematics (10th Ed.)
4. David C. Lay - Linear Algebra and Applications (4th Ed.)
5. Gilbert Strang - Linear Algebra and Applications (4th Ed.)

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ALL TOPICS COVERED - FORMULA-BASED SUBJECT! 📐