# VTU 1BMATM101 Differential Calculus and Linear Algebra: Study Guide

> Study guide for VTU 1BMATM101 (2025 scheme, Mechanical stream): polar curves, curvature, Taylor series, partial derivatives, first order ODEs and matrices.

Source: https://practicecalculus.com/courses/vtu-1bmatm101
Updated: 2026-11-08

## At a glance

- **Course code:** 1BMATM101 (Applied Mathematics-I for the Mechanical Engineering 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 tutorials)
- **Semester:** Semester 1
- **Assessment:** CIE 50 marks, SEE 50 marks, 3 hour theory exam
- **Prerequisites:** Not stated in the syllabus
- **Covers:** Polar curves and curvature, Taylor and Maclaurin series, indeterminate forms, partial differentiation, first order ODEs, matrices and eigenvalues
- **Next course:** [1BMATM201 Multivariable Calculus and Numerical Methods](/courses/vtu-1bmatm201) in Semester 2
- **Practice here:** [Calculus 1](/learn/calculus-1), [power series and polar](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) and [partial derivatives](/learn/calculus-3/partial-derivatives/functions-of-several-variables)

## Key takeaways

- 1BMATM101 is the Semester 1 Applied Mathematics-I course for the Mechanical Engineering stream in the VTU 2025 scheme.
- It carries 4 credits with 3 lecture hours and 2 tutorial hours a week, and is marked 50 CIE plus 50 SEE.
- Three of its five modules are calculus and differential equations: polar curves and curvature, series and multivariable calculus, and first order ODEs.
- The last two modules are linear algebra: rank, Gauss elimination, LU decomposition, Gauss-Seidel and eigenvalues.
- Other streams take a parallel course: 1BMATC101 (Civil), 1BMATE101 (Electrical) and 1BMATS101 (Computer Science).

**1BMATM101 Differential Calculus and Linear Algebra** is the Semester 1 mathematics course for the Mechanical Engineering stream at **Visvesvaraya Technological University (VTU)**, Belagavi, under the **2025 scheme** (effective from the academic year 2025-26). In the scheme it is the Applied Mathematics-I course, coded 1BMATx101, where the letter changes with your stream. This guide follows the official VTU syllabus PDF and maps every calculus topic to free practice.

## Course at a glance

The syllabus gives 4 credits with a 3:2:0:0 teaching pattern: 3 lecture hours and 2 tutorial hours a week, 40 hours of theory plus 20 hours of tutorials in total. Assessment is 50 marks of Continuous Internal Evaluation (CIE) and 50 marks in a 3 hour Semester End Examination (SEE). No prerequisite is stated, so the course assumes PUC or Class 12 mathematics.

| Stream | Course code | Title |
| --- | --- | --- |
| Civil | 1BMATC101 | Differential Calculus and Linear Algebra |
| Mechanical | 1BMATM101 | Differential Calculus and Linear Algebra |
| Electrical | 1BMATE101 | Differential Calculus and Linear Algebra |
| Computer Science | [1BMATS101](/courses/vtu-1bmats101) | Calculus and Linear Algebra |

## What 1BMATM101 covers

- **Module 1, Polar curves and curvature:** polar coordinates and polar curves, the angle between the radius vector and the tangent, the angle between two curves, pedal equations, and curvature and radius of curvature in Cartesian, parametric, polar and pedal forms.
- **Module 2, Series expansion, indeterminate forms and multivariable calculus:** Taylor's and Maclaurin's series in one variable, indeterminate forms and L'Hospital's rule, partial differentiation, the total derivative, differentiation of composite functions, Jacobians, and maxima and minima of functions of two variables.
- **Module 3, First order ODEs:** linear and Bernoulli equations, exact equations and equations reducible to exact form with an integrating factor, orthogonal trajectories, and the law of natural growth and decay.
- **Modules 4 and 5, Linear algebra:** row echelon form and rank, inverse by Jordan's method, Gauss elimination, LU decomposition, Gauss-Seidel, an application to traffic flow, eigenvalues and eigenvectors, Rayleigh's power method, diagonalisation, inverse by the Cayley-Hamilton theorem, and the Moore-Penrose pseudoinverse.

## 1BMATM101 topic map

PracticeCalculus focuses on calculus, so Modules 4 and 5 are not in the table.

| 1BMATM101 topic | Practice | Read |
| --- | --- | --- |
| 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) |
| Tangents to polar curves | [Polar slopes](/learn/calculus-2/power-series-parametric-polar/polar-slopes) | [Tangent function](/functions/tangent-function) |
| Curvature in parametric form | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) |
| Taylor's and Maclaurin's series | [Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |
| Indeterminate forms | [L'Hôpital's rule](/learn/calculus-1/applications-of-derivatives/lhopitals-rule) | [L'Hôpital's rule](/rules/lhopitals-rule) |
| Partial and total derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives) |
| 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) |
| Maxima and minima of two variables | [Second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Optimization practice](/practice-problems/optimization) |
| First order ODEs | [Separable equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Differential equations practice](/practice-problems/differential-equations) |
| Natural growth and decay | [Growth and decay](/learn/calculus-2/improper-integrals-and-applications/growth-and-decay) | [Exponential functions](/functions/exponential-functions) |

## Example problems at the 1BMATM101 level

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

**1. Indeterminate form.** $\displaystyle\lim_{x \to 0}\frac{1 - \cos x}{x^2} = \lim_{x \to 0}\frac{\sin x}{2x} = \frac{1}{2}$, applying L'Hospital's rule once and then the standard limit $\frac{\sin x}{x} \to 1$.

**2. Maclaurin series.** $\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ for $-1 < x \le 1$.

**3. A linear ODE.** Solve $y' + y = e^{x}$. The integrating factor is $e^{x}$, so $(ye^{x})' = e^{2x}$ and $y = \frac{1}{2}e^{x} + Ce^{-x}$.

**4. Radius of curvature.** For $y = x^2$ at the origin, $\rho = \frac{(1 + y'^2)^{3/2}}{|y''|} = \frac{1}{2}$.

## How to do well in 1BMATM101

- **Use all 20 tutorial hours.** The 3:2 pattern means almost half your contact time is problem solving. Bring attempted problems, not blank pages.
- **Learn the polar formulas with pictures.** Draw the radius vector and tangent each time; the angle formulas in Module 1 make sense once you see them.
- **Drill partial derivatives until they are automatic,** because Jacobians, total derivatives and maxima and minima all build on them. Try the [partial derivatives practice set](/practice-problems/partial-derivatives).
- **Practise in short daily sessions** on [practice](/practice), and look ahead to [1BMATM201](/courses/vtu-1bmatm201), which adds multiple integrals, higher order ODEs and vector calculus.

## Frequently asked questions

### What is 1BMATM101 in VTU?

1BMATM101 Differential Calculus and Linear Algebra is the first semester mathematics course for the Mechanical Engineering stream under the VTU 2025 scheme. It has 4 credits, a 3:2:0:0 teaching pattern, and is assessed with 50 marks of Continuous Internal Evaluation and a 50 mark, 3 hour Semester End Examination.

### Which modules are in 1BMATM101?

Module 1 is polar curves and curvature, Module 2 is series expansion, indeterminate forms and multivariable calculus, Module 3 is ordinary differential equations of first order, and Modules 4 and 5 are Linear Algebra 1 and 2.

### What is the difference between 1BMATM101 and 1BMATS101?

1BMATM101 is for the Mechanical stream and starts with polar curves, curvature and one variable series before partial derivatives and first order ODEs. 1BMATS101 Calculus and Linear Algebra is for the Computer Science stream and covers partial derivatives and vector calculus, followed by vector spaces and linear transformations.

### Which textbook does 1BMATM101 follow?

The main textbook is B. S. Grewal, Higher Engineering Mathematics (44th edition, 2021). The syllabus also lists Lipschutz and Lipson, Linear Algebra (Schaum's Outlines), Kreyszig, Advanced Engineering Mathematics, and Strang, Linear Algebra and its Applications.