'Multivariable Calculus' is a course offered in the second semester of  B. Tech. programs at the School of Engineering, Amrita Vishwa Vidyapeetham.


The course is expected to enable the students

  • To understand parameterisation of curves and to find arc length
  • To familiarise with calculus of multiple variables.
  • To use important theorems in vector calculus in practical problem

Course Outcomes

At the end of the course the student will be able to

CO1 S e lect suitable parameterization of curves and to find their arc lengths
CO2 Find partial derivatives of multivariable functions and to use the Jacobian in practical problems.
CO3 Apply Fundamental Theorem of Line Integrals, Green’s Theorem, Stokes’ Theorem, or Divergence Theorem to evaluate integrals.


/CO PO PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12
CO1 1 3 --- --- --- --- --- --- --- --- --- ---
CO2 1 2 --- --- 2 --- --- --- --- --- --- ---
CO3 2 2     3              

Course Evaluation Pattern:

  • Test-1 -25 marks (one hour test) after 15th lecture.
  • CA - 25 marks (Quizzes / assignments / lab practice)
  • Test – 2- 50 marks (two-hour test) at the end of 30th lecture.
  • Total - 100 marks.

Supplementary exam for this course will be conducted as a two-hour test for 100 marks.

Functions of severable variables: 5 hours

Functions, limit and continuity. Partial differentiations, total derivatives, differentiation of implicit functions and transformation of coordinates by Jacobian. Taylor’s series for two variables.

Vector Differentiation: 10 hours

Vector and Scalar Functions, Derivatives, Curves, Tangents, Arc Length, Curves in Mechanics, Velocity and Acceleration, Gradient of a Scalar Field, Directional Derivative, Divergence of a Vector Field, Curl of a Vector Field.

(Sections : 9.4, 9.5, 9.6, 9.9, 9.10, 9.11)

Vector Integration: 15 hours

Line Integral, Line Integrals Independent of Path. (Sections : 10.1, 10.2)

Green’s Theorem in the Plane, Surfaces for Surface Integrals, Surface Integrals, Triple Integrals – Gauss Divergence Theorem, Stoke’s Theorem. (Sections : 10.4, 10.5, 10.6, 10.7, 10.9 )

Lab Practice Problems

Graph of functions of two variables, shifting and scaling of graphs. Vector products. Visualizing different surfaces.

Text Book:

  1. Advanced Engineering Mathematics, E Kreyszig, John Wiley and Sons, Tenth Edition, 2018.

Reference Books:

  1. Advanced Engineering Mathematics by Dennis G. Zill and Michael R.Cullen, second edition, CBS Publishers, 2012.
  2. ‘Engineering Mathematics’, Srimanta Pal and Subhodh C Bhunia, John Wiley and Sons, 2012, Ninth Edition.
  3. ‘Calculus’, G.B. Thomas Pearson Education, 2009, Eleventh Edition.