Course

Unit 1

Vector Function and Space Curve: Vector-valued functions of a real variable – Algebraic operations, limits, derivatives and integrals. Parametric representation of space curve, tangent, principal normal and arc length of a space curve. Applications to curvilinear motion: velocity, speed and acceleration.  TB2 – Vol.1, Chapter 14: Section 14.1 to 14.10.

Partial Derivatives: Functions of several variables, limits and continuity, partial derivatives, higher order partial derivatives, Tangent planes and linear approximations, maximum and minimum values, Hessian matrix and its interpretation.  

TB3 – 14.1, 14.2, 14.3, 14.6, 14.7

Unit 2

Differential calculus of scalar and vector fields: Scalar point function and vector point function, scalar and vector fields. Examples: Euclidean distance field, velocity field, gravitational field and force field. Gradient of scalar field and its geometrical interpretation, level sets, gradient as surface normal, directional derivatives, conservative fields and potential functions. Divergence and Curl of vector field, physical interpretation, Laplacian of scalar field, vector identities involving gradient, divergence and curl.

TB1 – 9.4, 9.7, 9.8, 9.9

Unit 3

Double Integrals and Surface Integrals: Geometrical interpretation of double integrals, evaluation of double integrals change of order of integration, Jacobian and change of variables in double integrals. Parametric representation of surfaces, the fundamental vector product as a normal to the surface, surface and flux integrals with physical applications.

TB1 – 10.3, 10.5, 10.6

Unit 4

Fundamental Theorem on Vector Calculus: Line integrals, path independence of line integrals, Green’s theorem in the plane, Stokes theorem and Gauss Divergence theorem with geometrical and physical interpretations and applications.

TB1 – 10.1, 10.2, 10.4, 10.7, 10.9

Text Books / References

Suggested lab exercises:

Plot the vector valued function and visualize the space curve.

Find the velocity and acceleration vector for the parametric curve and plot them.

1. Plotting the scalar field and vector field.
2. Draw the level curves and level surfaces.
3. Find the gradient of the function and represent the gradient vector field graphically.
4. Find the Jacobian matrix and Jacobian determinant of the given transformation.
5. Find the divergence, curl and Laplacian of the given function.
6. Test whether the vector field is conservative or not.
7. Plot the surface of the function and visualize using 3D surface plot and contour plot.
8. Plot the function and observe the peaks and valleys.

Text Books:

1. Advanced Engineering Mathematics, Erwin Kreyszig, John Wiley and Sons, Tenth Edition, 2011.
2. Tom M. Apostol, Calculus, Volume 1, John Wiley & Sons, Second edition, 2007. George B. Thomas and Maurice D. Weir, Thomas’ Calculus, 14th Edition, Pearson Education, 2018.

This course introduces the fundamentals of multivariable calculus through hands-on learning using MATLAB. Topics include differential, integral and vector calculus for functions of more than one variable. These mathematical concepts are used extensively in engineering, physical sciences and in the machine learning optimization algorithms

Course Outcomes: After successful completion of the course, students will be able to

• CO1 : Understand and apply concepts of vector-valued functions, space curves, curvature and motion in space. Apply partial derivatives for tangent planes and Hessian matrix.
• CO2 : Analyze scalar and vector fields using gradient, divergence, curl and Laplacian.
• CO3 : Evaluate double integrals and surface integrals using Jacobian transformations.
• CO4 : Apply Green’s, Stokes’ and Gauss divergence theorems in solving physical and mathematical problems.

CO-PO Mapping: