Integral and Vector Calculus

Paper Code: 
CMAT 313
Credits: 
4
Contact Hours: 
60.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. Understand the basic concepts like indefinite and definite integrals, improper integrals real-valued functions of one variable (including algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions).
  2. Demonstrate the integral ideas of the functions defined including line, surface and volume integrals - both derivation and calculation in rectangular, cylindrical and spherical coordinate systems and understand the proofs of each instance of the fundamental theorem of calculus.
  3. Use the techniques of integration in several contexts, and to interpret the integral both as an anti-derivative and as a limit of a sum of products. 
  4. The basic concepts are illustrated by applying them to various problems where their application helps arrive at a solution.

 

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

CMAT 313

 

 

 

 

 

 

 

 

 

 

Integral and Vector Calculus

 (Theory)

 

 

 

 

 

 

 

The students will be able to –

 

CO41: Evaluate indefinite and definite integrals and use definite integrals to solve application problems.

CO42: Inter-relationship amongst the line integral, double and triple integral formulations.

CO43: Compute the derivatives and line integrals of vector functions and learn their applications

CO44: Applications of multivariable calculus tools in physics, economics, optimization, and
understanding the architecture of curves and surfaces in plane and space etc.

CO45: Evaluate surface and volume integrals and learn their inter-relations and applications.

CO46: Realize importance of Green, Gauss and Stokes’ theorems in other branches of
mathematics

Approach in teaching:

 

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Field trips

 

 

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

 

 

Unit I: 
I
9.00

Reduction formulae: (sin)^n x, (cos)^n x, (tan)^n x and (sin)^m x .(cos)^n x,where m, n are positive integers. Definition and properties of Gamma and Beta functions, Relation between Gamma and Beta functions, Duplication formula and problems related to these functions.

Unit II: 
II
9.00
Rectification: Length of cartesian and polar curves. Quadrature: Area of cartesian and polar curves, Volumes and surfaces of solids of revolution (cartesian and polar forms).
 
Unit III: 
III
9.00
Double integrals, Change of order of integration, Triple integrals, Dirichlet’s integral.
 
Unit IV: 
IV
9.00
Scalar and vector point function, Differentiation and integration of vector point function, Gradient, Directional derivatives, Divergence and curl of a vector point function.
 
Unit V: 
V
9.00
Identities involving differential vector operators, Gauss’ divergence, Stokes’ and Green’s theorems (without proof) their applications and related problems.
 
Essential Readings: 
  • Gorakh Prasad, A Text Book on Integral Calculus, Pothishala Pvt. Ltd, Allahabad, 2015.
  • Shanti Narayan, Integral Calculus, S. Chand & Co. Pvt. Ltd., New Delhi, 2005.
  • Shanti Narayan, A Text Book of Vector Calculus, S. Chand & Co. Pvt. Ltd. New Delhi, 2010.
  • M. Ray and H. S. Sharma, Vector Algebra and Calculus, Students and Friends Co. Agra, 1998.
 
References: 
  • Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley and Sons, 2008.
  • G. C. Sharma and Madhu Jain , Integral Calculus, Galgotia Publication, Dariyaganj, New Delhi,2000.
  • P. K. Mittal and Shanti Narayan, Integral Calculus, S.Chand & Co. Pvt. Ltd. New Delhi, 2005.
  • Muray R. Spiegel, Vector Analysis, Schaum Publishing Company, New York, 2007.
  • Saran and Nigam, Introduction to Vector Analysis, Pothisala Pvt. Ltd, Allahabad, 2001.  
 
Academic Year: 
2023-2024
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