VTU 1BMATS101 Calculus and Linear Algebra: Syllabus and Practice

1BMATS101 Calculus and Linear Algebra is the Semester 1 mathematics course for the Computer Science stream at Visvesvaraya Technological University (VTU) under the 2025 scheme, effective from the academic year 2025-26. It is the CSE version of Applied Mathematics-I (1BMATx101). Unlike the Mechanical, Civil and Electrical versions, it skips one variable topics and goes straight to multivariable and vector calculus, then spends three modules on linear algebra. This guide follows the official VTU syllabus PDF.

Course at a glance

The course outcomes ask you to apply multivariable and vector calculus to compute derivatives, optimise functions and analyse vector fields; to solve linear systems and find eigenvalues; to use vector spaces and linear transformations; and to demonstrate applications with modern ICT tools.

What 1BMATS101 covers

• Module 1, Calculus: partial differentiation, the total derivative, differentiation of composite functions, Jacobians, statement of Taylor's and Maclaurin's series for two variables, and maxima and minima of functions of two variables.
• Module 2, Vector calculus: scalar and vector fields, the gradient, directional derivatives, divergence and curl with physical interpretation, solenoidal and irrotational fields and scalar potentials, an introduction to polar coordinates and polar curves, and curvilinear coordinates (scale factors, base vectors, cylindrical and spherical polar coordinates, orthogonality).
• Module 3, Linear equations and eigenvalues: row echelon form and rank, Gauss elimination, Gauss-Jordan, traffic flow, eigenvalues, eigenvectors and diagonalisation.
• Module 4, Vector spaces and Module 5, Linear transformations: subspaces, span, independence, basis and dimension, inner products, the matrix of a linear map, and the rank-nullity theorem.

1BMATS101 topic map

Example problems at the 1BMATS101 level

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

1. A Jacobian. For x = r\cos\theta, y = r\sin\theta:

2. A directional derivative. For f = x^2 y at (1, 2) in the direction of \langle 3, 4 \rangle: \nabla f = \langle 2xy, x^2 \rangle = \langle 4, 1 \rangle, and D_u f = \frac{4 \cdot 3 + 1 \cdot 4}{5} = \frac{16}{5}.

3. Divergence and curl. For \mathbf{F} = \langle yz, xz, xy \rangle, \nabla \cdot \mathbf{F} = 0 and \nabla \times \mathbf{F} = \mathbf{0}, so \mathbf{F} is both solenoidal and irrotational, with scalar potential \phi = xyz.

How to do well in 1BMATS101

• Refresh one variable derivatives first. The course assumes them from day one. Ten minutes a day on derivative rules practice in the first week pays off quickly.
• Picture the vector fields. Divergence and curl are easier to remember when you can describe what the field does to a small paddle wheel or a small box.
• Use the tutorial hours. Two tutorial hours a week are part of the 3:2 pattern, so most of your problem practice can happen in class.
• Revise in short sessions on practice and keep the formula sheet handy for the SEE.