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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.

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.

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

Curvature and Radius of curvature in Cartesian form.

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

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

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 ****(****2****hr+0hr+****4****hr)**

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).

Problems on radius of curvature in Cartesian form

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

Vector algebra: Algebraic operations, Scalar product, Inner product

https://www.youtube.com/watch?v=fNk_zzaMoSs

https://www.youtube.com/watch?v=LyGKycYT2v0

https://www.slideshare.net/UrmilaBhardwaj/vector-algebra-58263932

Vector product, Scalar and vector triple product

https://www.youtube.com/watch?v=eu6i7WJeinw

https://www.slideshare.net/AshrafulTauhid/dot-cross-product-of-vectors

https://www.slideshare.net/guest581a478/triple-product-of-vectors-presentation

**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

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

Problems on Surface Integrals and Triple Integrals

https://www.youtube.com/watch?v=7iy83x8bv6o

https://www.slideshare.net/jigarsable/multiple-integraltripple-integral

Gauss Theorem

Application of Green’s theorem to evaluate the integrals

Application of Gauss theorem to evaluate the integrals

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

Stokes’ Theorem

Application of Stokes’ theorem to evaluate the integrals

Application of Stokes’ theorem to evaluate the integrals

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 […]

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