# Calculus

Teacher

Category

### Course Attendees

Still no participant

Still no reviews

# 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