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]