VTU 1BMATM201 Multivariable Calculus and Numerical Methods: Guide

1BMATM201 Multivariable Calculus and Numerical Methods is the Semester 2 mathematics course for the Mechanical Engineering stream at Visvesvaraya Technological University (VTU) under the 2025 scheme, effective from the academic year 2025-26. It is the Mechanical stream's Applied Mathematics-II course and follows 1BMATM101 from Semester 1. This guide uses the official VTU syllabus PDF and maps each calculus topic to free practice.

Course at a glance

The syllabus lists 4 credits with 3 lecture hours and 2 tutorial hours a week, for 40 hours of theory and 20 to 24 hours of tutorials. Assessment is 50 marks of Continuous Internal Evaluation and 50 marks in a 3 hour theory Semester End Examination. The course outcomes ask you to model area, volume, heat conduction and field problems with integral calculus, higher order ODEs and vector calculus; to apply numerical methods to equations, interpolation, integration and ODEs; and to show mechanical engineering applications using modern ICT tools.

What 1BMATM201 covers

• Module 1, Integral calculus: definition and evaluation of double and triple integrals, change of order of integration, changing to polar coordinates, area and volume by double integrals, and beta and gamma functions with the relation between them.
• Module 2, Higher order ODEs: linear equations with constant coefficients, homogeneous and non homogeneous, with right hand sides e^{ax}, \sin(ax + b), \cos(ax + b) and x^n; variation of parameters; Cauchy's and Legendre's equations; and the mass spring model.
• Module 3, Vector calculus: scalar and vector fields, gradient, directional derivative, divergence and curl, solenoidal and irrotational fields and scalar potentials, line integrals, work done and flux, and statements of Green's and Stokes' theorems with problems.
• Module 4, Numerical methods 1: Regula-Falsi and Newton-Raphson, finite differences, Newton's forward and backward formulas, Newton's divided difference and Lagrange interpolation, and the trapezoidal, Simpson's 1/3 and Simpson's 3/8 rules.
• Module 5, Numerical methods 2: Taylor's series method, modified Euler's method, fourth order Runge-Kutta, and Milne's and Adams-Bashforth predictor-corrector methods.

1BMATM201 topic map

Example problems at the 1BMATM201 level

These are original practice problems, not VTU question paper items.

1. A gamma function. \Gamma(n) = \int_0^\infty e^{-x}x^{n-1}\,dx, and \Gamma(n + 1) = n\Gamma(n), so \int_0^\infty x^4 e^{-x}\,dx = \Gamma(5) = 4! = 24.

2. Green's theorem. For \oint_C (x\,dy - y\,dx) around the unit circle,

3. Simpson's 1/3 rule. With h = 0.5 on [0, 1], \int_0^1 x^2\,dx \approx \frac{0.5}{3}(0 + 4(0.25) + 1) = \frac{1}{3}, which is exact because Simpson's rule integrates cubics exactly.

4. Change to polar. \iint_{x^2 + y^2 \le 4} e^{-(x^2 + y^2)}\,dA = \int_0^{2\pi}\int_0^2 e^{-r^2} r\,dr\,d\theta = \pi(1 - e^{-4}).

How to do well in 1BMATM201

• Sketch every region. In Module 1 the limits decide everything, so draw the region before you write the integral, especially for change of order.
• Keep the 1BMATM101 skills sharp. Partial derivatives come back as gradients, divergence and curl in Module 3.
• Set up numerical tables neatly. Interpolation and Runge-Kutta questions are long; a clean table prevents copying errors.
• Practise in short daily sessions on practice, and use the formula sheet to memorise the vector identities before the SEE.