BMATS101 - MATHEMATICS-I FOR CSE STREAM
MODULE 1: CALCULUS (8 Hours)
Topics:
-
Polar Coordinates
- Polar curves
- Angle between radius vector and tangent
- Angle between two curves
-
Pedal Equations
-
Curvature and Radius of Curvature
- Cartesian form
- Parametric form
- Polar form
- Pedal form
Self-Study:
- Center and circle of curvature
- Evolutes and involutes
MODULE 2: SERIES EXPANSION AND MULTIVARIABLE CALCULUS (8 Hours)
Topics:
-
Series Expansion
- Taylor's series (Statement only)
- Maclaurin's series (Statement only)
- Problems
-
Indeterminate Forms
- L'Hospital's rule
- Problems
-
Partial Differentiation
- Total derivative
- Differentiation of composite functions
- Jacobian
- Problems
-
Maxima and Minima
- For functions of two variables
- Problems
Self-Study:
- Euler's theorem
- Method of Lagrange's undetermined multipliers (single constraint)
MODULE 3: ORDINARY DIFFERENTIAL EQUATIONS (ODEs) OF FIRST ORDER (8 Hours)
Topics:
-
Linear and Bernoulli's Differential Equations
-
Exact Differential Equations
- Reducible to exact
- Integrating factors
-
Applications
- Orthogonal trajectories
- L-R circuits
- C-R circuits
-
Non-linear Differential Equations
- General and singular solutions
- Solvable for p only
- Clairaut's equations
- Reducible to Clairaut's equations
Self-Study:
- Applications of ODEs
- Rate of Growth or Decay
- Conduction of heat
- Solvable for x and y
MODULE 4: MODULAR ARITHMETIC (8 Hours)
Topics:
-
Congruences
- Introduction
- Linear congruences
- The Remainder theorem
-
Solving Polynomials
-
Linear Diophantine Equation
-
System of Linear Congruences
-
Theorems
- Euler's Theorem
- Wilson's Theorem
- Fermat's Little Theorem
-
Applications
- RSA algorithm
- Cryptography
- Encoding and decoding
- Public key encryption
Self-Study:
- Divisibility
- GCD
- Properties of Prime Numbers
- Fundamental theorem of Arithmetic
MODULE 5: LINEAR ALGEBRA (8 Hours)
Topics:
-
Matrix Operations
- Elementary row transformations
- Rank of a matrix
-
System of Linear Equations
- Consistency test
- Gauss-elimination method
- Gauss-Jordan method
- Gauss-Seidel method (approximate solution)
-
Eigenvalues and Eigenvectors
- Computation
- Rayleigh's power method
- Dominant eigenvalue and eigenvector
Self-Study:
- Gauss-Jacobi iterative method
- Inverse of matrix by Cayley-Hamilton theorem
Applications:
- Boolean matrix
- Network Analysis
- Markov Analysis
- Critical point of network system
- Optimum solution
LABORATORY EXPERIMENTS (10 Sessions)
- 2D plots for Cartesian and polar curves
- Angle between polar curves, curvature and radius of curvature
- Partial derivatives and Jacobian
- Maxima and Minima of two variables
- First-order ODE solution and plotting curves
- GCD using Euclid's Algorithm
- Solving linear congruences ax ≡ b (mod m)
- System of linear equations (consistency + graphical representation)
- Gauss-Seidel iteration
- Eigenvalues, eigenvectors and Rayleigh power method
Software: Mathematica/MATLAB/Python/Scilab
KEY FORMULAS & CONCEPTS:
Module 1:
- tan φ = r/(dr/dθ)
- Radius of curvature formulas (all forms)
Module 2:
- Taylor series
- Maclaurin series
- L'Hospital's rule: lim[f(x)/g(x)] = lim[f'(x)/g'(x)]
- Jacobian: J = ∂(u,v)/∂(x,y)
Module 3:
- Linear DE: dy/dx + Py = Q
- Bernoulli's DE: dy/dx + Py = Qyⁿ
- Exact condition: ∂M/∂y = ∂N/∂x
- Clairaut's equation: y = px + f(p)
Module 4:
- a ≡ b (mod m)
- Euler's theorem: a^φ(m) ≡ 1 (mod m)
- Fermat's little theorem: a^(p-1) ≡ 1 (mod p)
- Wilson's theorem: (p-1)! ≡ -1 (mod p)
Module 5:
- Gauss elimination
- Gauss-Seidel: x_i^(k+1) = (1/a_ii)[b_i - Σa_ij x_j]
- Eigenvalue equation: Av = λv
- Power method for dominant eigenvalue
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)
TEXTBOOKS:
- B.S. Grewal - Higher Engineering Mathematics (44th Ed.)
- E. Kreyszig - Advanced Engineering Mathematics (10th Ed.)
- David M Burton - Elementary Number Theory (7th Ed.)
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