# VTU 1BMATS101 Calculus and Linear Algebra: Syllabus and Practice

> Study guide for VTU 1BMATS101 (2025 scheme, CSE stream): partial derivatives, Taylor series in two variables, vector calculus, matrices and vector spaces.

Source: https://practicecalculus.com/courses/vtu-1bmats101
Updated: 2026-11-08

## At a glance

- **Course code:** 1BMATS101 (Applied Mathematics-I for the Computer Science stream)
- **Scheme:** 2025 scheme, effective from the academic year 2025-26
- **Credits:** 4 credits (L:T:P:S 3:2:0:0, 40 hours theory + 20 hours tutorial)
- **Semester:** Semester 1
- **Assessment:** CIE 50 marks, SEE 50 marks, 3 hour theory exam
- **Prerequisites:** Not stated in the syllabus
- **Covers:** Partial differentiation, Jacobians, two variable Taylor series, maxima and minima, vector calculus, curvilinear coordinates, linear equations, eigenvalues, vector spaces and linear transformations
- **Next course:** 1BMATS201 Numerical Methods in Semester 2
- **Practice here:** [Partial derivatives](/learn/calculus-3/partial-derivatives/functions-of-several-variables) and [vector calculus](/learn/calculus-3/vector-calculus/vector-fields) in Calculus 3

## Key takeaways

- 1BMATS101 is the Semester 1 Applied Mathematics-I course for the Computer Science stream in the VTU 2025 scheme.
- It carries 4 credits with a 3:2:0:0 pattern and is marked 50 CIE plus 50 SEE.
- The first two modules are multivariable and vector calculus; the last three are linear algebra.
- It goes straight to functions of several variables, so one variable differentiation from PUC must already be solid.
- In Semester 2, CSE stream students take 1BMATS201 Numerical Methods.

**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

| Item | Detail |
| --- | --- |
| Credits | 4 |
| Teaching hours a week (L:T:P:S) | 3:2:0:0 |
| Total hours | 40 theory + 20 tutorial |
| Assessment | CIE 50 + SEE 50 = 100 marks |
| SEE duration | 3 hours, theory |
| Main textbook | B. S. Grewal, Higher Engineering Mathematics, 44th edition |

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

| 1BMATS101 topic | Practice | Read |
| --- | --- | --- |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial differentiation | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Total derivative and composite functions | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule](/rules/chain-rule) |
| Jacobians | [Change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) | [Partial derivatives practice](/practice-problems/partial-derivatives) |
| Taylor series in two variables | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Taylor series](/blog/taylor-series) |
| Maxima and minima | [Second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Optimization practice](/practice-problems/optimization) |
| Gradient and directional derivatives | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Scalar and vector fields | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) | [Vectors, dot and cross product](/blog/vectors-dot-and-cross-product) |
| Divergence and curl | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | [Divergence theorem](/theorems/divergence-theorem) |
| Irrotational fields and scalar potential | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) |
| Polar coordinates and polar curves | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) | [Parametric and polar practice](/practice-problems/parametric-and-polar) |
| Cylindrical and spherical coordinates | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Points and distance](/learn/calculus-3/vectors-and-space/points-and-distance) |

## Example problems at the 1BMATS101 level

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

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

$$\frac{\partial(x, y)}{\partial(r, \theta)} = x_r y_\theta - x_\theta y_r = \cos\theta \cdot r\cos\theta - (-r\sin\theta)\sin\theta = r$$

**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](/learn/calculus-1/derivative-rules/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](/practice) and keep the [formula sheet](/formulas) handy for the SEE.

## Frequently asked questions

### What is 1BMATS101 in VTU?

1BMATS101 Calculus and Linear Algebra is the first semester mathematics course for the Computer Science stream in the VTU 2025 scheme. It is a 4 credit course with 3 lecture and 2 tutorial hours a week, assessed through 50 marks of CIE and a 50 mark, 3 hour SEE.

### Which modules are in 1BMATS101?

Module 1 is calculus (partial differentiation, total derivative, composite functions, Jacobians, Taylor and Maclaurin series in two variables, maxima and minima). Module 2 is vector calculus with polar and curvilinear coordinates. Module 3 covers linear equations, eigenvalues and eigenvectors, Module 4 vector spaces, and Module 5 linear transformations.

### Does 1BMATS101 include integration?

No. The 1BMATS101 syllabus has no integration module. Its calculus content is differential: partial derivatives, gradient, divergence and curl.

### Which books are prescribed for 1BMATS101?

The textbooks are B. S. Grewal, Higher Engineering Mathematics (44th edition), Kreyszig, Advanced Engineering Mathematics (10th edition), and Lipschutz and Lipson, Linear Algebra (Schaum's Outlines). References include Ramana, Strang, Bali and Goyal, Stewart's Calculus, Poole, Lay and Williams.