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# Code(Credit) : CUTM1925(2-0-1)

## Course Objectives

To apply the concepts of derivative to find curvature and radius of curvature of a curve.

To apply concepts of Vector Calculus to the problems related to models in work, circulation and flux Problems, hydrodynamics and fluid dynamics etc.

## Learning Outcomes

Upon successful completion of this course, students will be able to:

Calculate curvature and radius of curvature for a given curve.

Determine the important quantities associated with scalar and vector fields.

Find gradient of a scalar point function, divergence and curl of a vector point function.

Evaluate line integral, double integral and applying these concepts to find out work done by a force, volume of regions in space, center of gravity of a mass etc.

Transform double integral to line integrals, triple integrals to surface integrals, surface integrals to line integrals and vice versa.

## Course Syllabus

Module-I (3hr+0hr+2hr)

Curvature and Radius of curvature in Cartesian form.

Project 1: To find radius of curvature (Parametric form)

Module-II (2hr+0hr+4hr)

Vector algebra: Algebraic operations, Scalar product, Inner product, Vector product, Scalar and vector triple product.

Project 2: Problems based on inner product, scalar and vector triple products.

Project 3: To find angle between two vectors, area of triangle and parallelogram, volume of parallelepiped and tetrahedron using vector algebra.

Module III (2hr+0hr+4hr)

Gradient of scalar point function, Directional derivatives, Divergence and curl of vector point functions, second order differential operator: the Laplacian operator.

Project 4: To prove the identities with regards to Gradient, Divergence and Curl.

Project 5: To find normal vector to a plane using Gradient of scalar point function.

Module-IV: (3hr+0hr+0hr)

Line Integrals (path dependence and path independence), double integrals.

Module-V: (3hr+0hr+0hr)

Surface Integrals, Triple Integrals

Module-VI: (4hr+0hr+2hr)

Green’s and Gauss’s Theorems (without proof) and their applications to evaluate the integrals.

Project 6: To find center of gravity and moments of inertia of a mass density

Module-VII: (3hr+0hr+0hr)

Stokes’ Theorem (without proof) and its applications to evaluate the integrals.

Text Books:

1. A Text book of Calculus Part – II by Shanti Narayan, Publisher: S. Chand & Company Ltd.

Chapters: 8 (Art. 24, 25 (only for Cartesian and parametric curves)).

2. Advanced Engineering Mathematics by E. Kreyszig, Publisher: John Willey & Sons Inc.- 8th Edition

Chapters: 8 (8.1 to 8.3, 8.9 to 8.11), 9 (9.1 to 9.7, 9.9).

## Session 3

Problems on radius of curvature in Cartesian form

## Session 4&5

Project 1: To find radius of curvature (Parametric form)

## Session 8&9

Project 2: Problems based on inner product, scalar and vector triple products

## Session 10&11

Project 3: To find angle between two vectors, area of triangle and parallelogram, volume of parallelepiped and tetrahedron using vector algebra

## Session 12

Gradient of scalar point function, Directional derivatives

## Session 13

Divergence and curl of vector point functions, second order differential operator: the Laplacian operator

https://www.slideshare.net/hdiwakar/vector-calculus-1st-2

## Session 14&15

Project 4: To prove the identities with regards to Gradient, Divergence and Curl

## Session 16&17

Project 5: To find normal vector to a plane using Gradient of scalar point function

## Session 18

Line Integrals (path dependence)

## Session 19

Line Integrals (path independence)

Gauss Theorem

## Session 26

Application of Green’s theorem to evaluate the integrals

## Session 27

Application of Gauss theorem to evaluate the integrals

## Session 28&29

Project 6: To find center of gravity and moments of inertia of a mass density

Stokes’ Theorem

## Session 31

Application of Stokes’ theorem to evaluate the integrals

## Session 32

Application of Stokes’ theorem to evaluate the integrals

## Case Studies

### Our Main Teachers ### Dr. Goutam Kumar Mahato

##### Teaching
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Received M.Sc. (Mathematics) from Vinoba Bhave University, Hazaribag and B.Ed. from Ranchi University, Ranchi in the year of 2006 and 2008 respectively. Awarded PhD, in the year of 2013, from Indian School of Mines, Dhanbad. Published 22 research papers in different journals of international repute. Participated and presented 16 research papers in various National and […]