Mathematics I
MA1001E MATHEMATICS I
| L | T | P | O | C |
|---|---|---|---|---|
| 3 | 1* | 0 | 5 | 3 |
Total Lecture Sessions: 39
Course Outcomes:
- CO1: Formulate some engineering problems as ODEs and hence solve such problems.
- CO2: Solve linear ODEs with constant coefficients.
- CO3: Find the limits, check for continuity and differentiability of real valued functions of two variables.
- CO4: Test for the convergence of sequences and series.
- CO5: Find the Fourier series representing periodic functions.
Course Content:
Ordinary Differential Equations
- Existence and uniqueness of solution of first order ODE
- Methods of solutions of first order ODE
- Linear ODE
- Orthogonal Trajectories
- Linear homogeneous second order ODEs with constant coefficients
- Fundamental system of solutions
- Existence and uniqueness of solutions
- Wronskian
- Method of undetermined coefficients
- Solution by variation of parameters
- Euler-Cauchy equations
- Applications of first and second order ODEs
- System of linear ODEs with constant coefficients
Functions of Several Variables
- Limit, Continuity
- Partial derivatives, Partial differentiation of composite functions
- Directional derivatives, Gradient
- Local maxima and local minima of functions of two variables
- Critical point, Saddle point
- Taylor’s formula for two variables, Hessian, Second derivative test
- Method of Lagrange multipliers
- Parameterised curves in space, Arc length, Tangent and normal vectors, Curvature and torsion
Sequences, Series, and Fourier Analysis
- Sequences, Cauchy sequence, Convergence of sequences
- Series, Convergence of series, Tests for convergence, Absolute convergence
- Sequence of functions, Power series, Radius of convergence, Taylor series
- Periodic functions and Fourier series expansions, Half-range expansions
- Fourier integral, Fourier transforms and their properties
References:
- 1. Anton, H., Bivens, I., & Davis, S. (2015). *Calculus*, 10th Ed. New York: John Wiley & Sons. [1]
- 2. Thomas, G. B., Weir, M. D., & Hass, J. (2015). *Thomas' Calculus*, 12th Ed. New Delhi, India: Pearson Education. [2]
- 3. Kreyszig, E. (2015). *Advanced Engineering Mathematics*, 10th Ed. New York: John Wiley & Sons. [3]
- 4. Apostol, T. M. (2014). *Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra*, 1st Ed. New Delhi: Wiley. [4]