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- To study how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models.
- To construct a relatively simple quantitative models of change, and to deduce their consequences.

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

- understand the importance of linear functions in mathematics.
- deals with the major problems of differential and integral calculus.
- recognize other important classes of functions (such as trigonometric and rational functions), and will be able to use calculus with these functions.

**Module-I: ****Curvature**

Curvature, circle of curvature and radius of curvature, Radius of curvature in Cartesian and polar form.

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

**Project 2:** To find radius of curvature (Pedal form)

**Module-II: ****Asymptotes**

Asymptotes and determination of asymptotes, Asymptotes parallel to coordinate axis, Rules to find asymptotes.

**Project 3:** To find asymptotes of different curves in implicit form

**Module-III: ****Curve Tracing**

Curve Tracing, Tracing of Cartesian, parametric and polar curves.

**Project 4:** To trace some Cartesian curves (Folium of Descartes, Astroid)

**Project 5:** To trace some polar curves (Cardioid, Lemniscate of Bernaulli)

**Project 6:** To trace few more curves (Cartenary, Cycloid etc) and Loops

**Practice 1: **Plotting the graphs of the functions e^{ax+b}, log (ax + b), 1/(ax + b), sin (ax + b), cos(ax + b), |ax +b| and to illustrate the effect of a and b on the graph

**Practice 2: **Plotting the graphs of the polynomial of degree 4 and 5, the derivative graph, the second derivative graph and comparing them

**Practice 3: **Sketching parametric curves (e.g. Trochoid, cycloid, epicycloids, hypocycloid)

**Module-IV: ****Reduction Formula**

Reduction formula, Reduction formula involving sine and cosine function sin ^nx cos ^nx, Reduction formula for integrals with limits from 0 to pi/2, Reduction formula for integral of sin^px, Cos^qx and also with limits from 0 to Pi/2.

**Project 7:** To find the Reduction formula for integrals of tan^nx, Cot ^nx, sec^nx, cosec^nx

**Project 8:** To find the Reduction formula for integral of sin^pxcos^qx for different values of p and q and different cases

**Module-V: ****Rectification and Quadrature**

Rectification of curves, Finding arc length for polar curves, Quadrature and areas of plane regions for Cartesian, parametric and polar curves, Sectorial area.

**Project 9: **To find the Area bounded by closed curves

**Module-VI: ****Volume of Solid Body Revolution**

Volume of solid of revolution, Volume of solid of revolution about any axis, Pappus theorem for volume of revolution.

**Practice 4:** Tracing of conics in Cartesian coordinates/polar coordinates

**Practice 5:** Sketching ellipsoid, hyperboloid of one and two sheets,

**Practice 6:** Sketching elliptic cone, elliptic paraboloid, hyperbolic paraboloid using Cartesian coordinates

** **** **

**Module-VII: ****Vector Differential Calculus**

Introduction to vector calculus: Definitions and characteristics of vectors, algebraic operations, Directional derivatives and directional derivatives along any line, Gradient of scalar point function, Divergence and curl of vector point functions, Second order differential operator the Laplacian operator and its properties.

**Project 10:** To explain about vector, different types of vector and vector algebra with suitable example

**Project 11:** To explain the geometrical significance of Gradient of scalar point function with some examples

**Project 12:** To discuss Laplacian operator and its different properties

**Text Books:**

1 A Text book of Calculus Part – II : Shanti Narayan

Chapters: 8 (Art. 24, 25, 26), 10 (Art. 33, 34, 35, 36, 37, 38).

2 A Text book of Calculus Part-III : Shanti Narayan

Chapters: 1 (Art 1, 2, 3), 3(Art 7, 8, 9), 4(Art 10, 11, 12) omitting Simpson’s rule), 5(Arts 13, 14), 6(Arts 15, 16).

3 A Textbook of Vector Calculus by Shanti Narayan & P. K. Mittal, S. Chand & Co., 2003

Chapters: 1, 6, 7 (7.1 to 7.6, 7.8 & 7.11)

**Reference Books**:

1 J. Strauss, G.L. Bradley and K. J. Smith, Calculus, 3rd Ed., Dorling Kindersley (India) P. Ltd. (Pearson Education). Delhi, 2007.

Chapters: 4(4.3, 4.4, 4.5 & 4.7), 9(9.4), 10(10.1-10.4).

2 Anton, I. Bivens and S. Davis, Calculus, 7th Ed., John Wiley and Sons (Asia) P. Ltd., Singapore, 2002.

Chapters: 6. (6.2-6.5).7(7.8). 8(8.2-8.3, Pages: 532-538), 11(11.1), 13(13.5)

3 B. Thomas and R. L. Finney. Calculus, 9th Ed., Pearson Education, Delhi, 2005

4 Courant and F. John Introduction to Calculus and Analysis (Volumes I & II). Springer-Verlag. New York. Inc., 1989

** **

To find radius of curvature (Intrinsic and Parametric form)

To find radius of curvature (Pedal form)

Asymptotes and determination of asymptotes

https://www.youtube.com/watch?v=oBmnt5tXRws

Asymptotes parallel to coordinate axis

https://www.youtube.com/watch?v=LCfxTH1LEd4

https://slideplayer.com/slide/5279710/

To find asymptotes of different curves (implicit form)

To trace some Cartesian curves (Folium of Descartes, Astroid)

To trace some polar curves (Cardioid, Lemniscate of Bernaulli)

To trace few more curves (Cartenary, Cycloid etc) and Loops

Plotting the graphs of the functions e^{ax+b}, log (ax + b), 1/(ax + b), sin (ax + b), cos(ax + b), |ax +b| and to illustrate the effect of a and b on the graph

Plotting the graphs of the polynomial of degree 4 and 5, the derivative graph, the second derivative graph and comparing them

Sketching parametric curves (e.g. Trochoid, cycloid, epicycloids, hypocycloid)

Reduction formula involving sine and cosine function sin ^nx cos ^nx

Reduction formula for integrals with limits from 0 to pi/2

Reduction formula for integral of sin^px, Cos^qx and also with limits from 0 to Pi/2

Reduction formula for integrals of tan^nx, Cot ^nx, sec^nx, cosec^nx

To find the Reduction formula for integral of sin^pxcos^qx for different values of p and q and different cases

Quadrature and areas of plane regions for Cartesian curve

https://www.youtube.com/watch?v=xec6HTcn2M8

https://www.youtube.com/watch?v=cHk_BKIoIXM&feature=youtu.be

https://www.slideshare.net/vaibhavtandel4/application-of-definite-integrals

Area of plane regions for polar curve

To find the area bounded by closed curves

Tracing of conics in Cartesian coordinates/polar coordinates

Sketching ellipsoid, hyperboloid of one and two sheets

Sketching elliptic, cone, elliptic, paraboloid, hyperbolic paraboloid using Cartesian coordinates

Introduction to vector calculus: Definitions and Characteristics of vectors

https://www.youtube.com/watch?v=KBSCMTYaH1s

https://www.slideshare.net/ManmohanDash/engineering-physics-vector-calculus

Algebraic operations on vectors

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

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

Directional derivatives and directional derivatives along any line

Second order differential operator

To explain about vector, different types of vector and vector algebra with suitable example

To explain the geometrical significance of Gradient of scalar point function with some examples

To discuss Laplacian operator and its different properties

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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