# IDC111- Mathematical Tools I

**Lecture Hours:**

Wednesday - 10.45-11.45am

Friday: 9.00 - 10 am.

**Practicals: (Mathematica)**

Group 1: Monday 2:15-4:15pm

Group 2: Monday 4.15-6.15pm

Group 3: Wednesday 2:15-4.15pm

Group 4: Wednesday 4.15-6.15pm

**Teaching Assistants:**

Prahlad Kanti Barman,

Vaisakh C P,

Arjun Unnikrishnan,

Lakshmi KP,

Gayathri V

Ashby Philip

Each student will be assigned a TA; the students are free to contact the TA's personally, through e-mail or through WhatsApp for any help regarding the course. The contact details will be shared with the students)

**Course structure:**

**Lectures:**

**Module 1: Vector Analysis- Part1:**

*(7 lecture/ tutorial sessions: 10-31 August 2016)*

Review of vector algebra: addition, subtraction and product of two vectors-polar and axial vectors with examples; triple and quadruple product. Concept of Scalar and Vector ﬁelds. Differentiation of a vector w.r.t. a scalar unit tangent vector and unit normal vector. Directional derivatives - gradient, divergence, curl and Laplacian operations and their meaning.

**Mid-Semester Exam I :- 1-3 Sept 2016 Weightage - 15%**

**Module 2: Vector Analysis- Part 2:**

*(6 lecture/ tutorial sessions: 7- 30 Sept 2016)*

Concept of line, surface and volume integrals. Statement of Gauss’ and Stokes’ theorems with physical examples. Gradient, divergence and curl in spherical polar and cylindrical coordinate systems.

**Mid-Semester Exam II :- 3-5 Oct 2016 Weightage - 15%**

**Module 3: Fourier Series:**

*(7 lecture/ tutorial sessions: 7 Oct - 4 Nov 2016)*

Fourier expansion of a periodic functions. Fourier Integrals

**Module 4: Complex numbers and functions:**

*(2 lecture/ tutorial sessions: 9-11 Nov 2016)*

Arithmetic operation, conjugates, modulus, polar form, powers and roots; Derivative;

** Final Exam :- 15--30 Nov 2016 Weightage (Theory) - 35%**

**References:**

Murray R. Spiegel, Schaum’s Outlines Vector Analysis, Tata Mcgraw Hill 2009.

E. Kreyszig, Advanced Engineering Mathematics, 8th Edition Wiley India Pvt Ltd, 2006.

**Mathematica Exercises: Weightage - 20%**

**Module 1 (2** sessions): Introduction to MATHEMATICA. Importing/exporting formatted datasets. Plotting of functions and data in 2D, 3D; Plotting parametrically deﬁned functions. Basic mathematical operations; symbolic differentiation of single and multi variable functions. Simple data ﬁtting (e.g. polynomial, exponential functions etc), error estimation. Examples for Taylor series expansion, demonstration of convergence. Programming in MATHEMATICA, debugging and execution.

**Module 2** (3 sessions): Plotting vectors in 3D; algebraic operations, span and linear independence. Visualizing the plane determined by two vectors; determining the unit normal from vector product. Obtaining equation of the plane and parametric representation of the same. Plotting a system of simple contours and surfaces as a visual representation of scalar ﬁelds. Determining the gradient of a scalar ﬁeld and graphical representation of the gradient as vectors. Visualization of various types of vector ﬁelds (divergent, rotational etc.) in 2D and 3D. Determination of divergence and curl of vector ﬁelds and their graphical representation. Real life scalar (temperature) and vector ﬁelds (static and rotating garden sprinkler, liquid vortex) and practical applications of the gradient, divergence and curl.

**Module 3** (1 sessions): Demonstration of Fourier series representation for simple waveforms (e.g. Square, triangular, saw tooth).

**Module 4** (1 session): Algebraic Manipulation of complex functions.

**Tentative Plan**

Exercise 1: 17/20 August 2016

Exercise 2: 22/24 August 2016

Exercise 3: 29/31 August 2016

Exercise 4: 5/7 Sept 2016

Exercise 5: 19/21 Sept 2016

Exercise 6: 26/28 Sept 2016

Exercise 7: 17/19 Oct 2016

Exercise 8: 24/26 Oct 2016

Exercise 9: 31 Oct/2 Nov 2016

Exercise 10: 7/9 Nov 2016

** Final Exam :- 15--30 Nov 2016 Weightage (Mathematica) - 10%**

**Reference:**

Stephen Wolfram, The MATHEMATICA Book, 5th Edition.

**Theory Assignments: Weightage 5%**