**JNTUK Mathematics – III Material/Notes **

Students enrolled in JNTUK R20 CSE, ECE, or Civil branches may be interested in downloading unit-wise materials for Mathematics III, which covers topics such as vector calculus, transforms, and partial differential equations (PDE-M3). These materials, which may be PDF notes, can help review and reinforce the concepts covered in this course. You can find these materials online or ask your professors or classmates for recommendations.”

**Course Objectives:**

- Developing an understanding of advanced mathematical concepts and techniques, such as vector calculus, transforms, and partial differential equations
- Improving problem-solving skills through the application of mathematical concepts and techniques to real-world situations
- Enhancing analytical and critical thinking skills through the analysis and synthesis of mathematical ideas
- Developing the ability to communicate mathematical concepts and ideas effectively, both in written and oral form
- We prepare students for further study in mathematics or related fields by providing a solid foundation in advanced mathematical concepts and techniques.

Consult the course syllabus or speak with your instructor to get a more specific list of objectives for your Mathematics III course.

**UNIT-1**

Vector calculus is a branch of mathematics that studies vector fields and how they change and interact. Some of the main concepts in vector calculus include the following:

- Vector differentiation: This refers to the process of taking the derivative of a vector field, which involves calculating the rate of change of the area at a given point. Some key concepts in vector differentiation include the gradient, directional derivative, divergence, and curl.
- Vector integration refers to finding a region’s area, volume, or another measure using vectors rather than scalar quantities. Some key concepts in vector integration include line integrals, work done, and surface and volume integrals. Several vector integral theorems, such as Greens, Stokes, and Gauss Divergence theorems, provide valuable tools for solving problems involving vector fields.

**Engineering Mathematics 3 Notes – UNIT-1**

**UNIT-2**

In Laplace transforms, the Laplace transform of a function is calculated by taking the integral of the process concerning time, with the integration being from 0 to infinity. Laplace transforms have several valuable properties, including the ability to transform derivatives and integrals of functions into algebraic equations, the ability to shift tasks in the time domain, and the ability to use convolution to find the transform of the convolution of two functions. These properties can be used to solve ordinary differential equations, particularly initial value problems, which involve finding the solution to a differential equation subject to specified initial conditions. Some key concepts in the Laplace transform include the Laplace transform of standard functions, shifting theorems, transforms of derivatives and integrals, the unit step function and Dirac’s delta function, inverse Laplace transforms, and the convolution theorem.

**Engineering Mathematics 3 Notes – UNIT-2**

**UNIT-3**

Fourier series and Fourier transforms are mathematical techniques used to represent periodic functions and functions of a continuous variable, respectively, as infinite series of sines and cosines. Fourier series represents a regular function as a sum of sines and cosines of different frequencies and amplitudes. Fourier transforms define a function as an integral of sines and cosines of different frequencies and amplitudes. Fourier series and Fourier transforms have several functional properties and applications, including the ability to extract the frequency components of a function and solve differential equations.

**Engineering Mathematics 3 Notes – UNIT-3**

**UNIT-4**

Partial differential equations (PDEs) are mathematical equations that involve partial derivatives and are used to describe how a function depends on multiple variables. PDEs of first order involve only first-order partial derivatives and can be formed by eliminating arbitrary constants and processes from a given problem. There are two main types of PDEs of the first order: linear and nonlinear. Linear PDEs can be solved using various methods, while nonlinear PDEs may require special techniques or approximations. Standard types of nonlinear PDEs include quasilinear, semilinear, and fully nonlinear.

**Mathematics 3 Book Pdf – UNIT-4**

**UNIT-5**

Second-order partial differential equations (PDEs) involve second-order partial derivatives. They can be solved using methods such as separating variables and eigenfunction expansion. PDEs have many applications, including the solution of one-dimensional wave and heat equations and the two-dimensional Laplace equation. The method of separation of variables is a common technique for solving these equations, which involves expressing the unknown function as the product of two or more processes and then solving the resulting system of ordinary differential equations.

**Mathematics 3 Book Pdf – UNIT-5**

**Engineering Mathematics 3 Textbook :**

- “Higher Engineering Mathematics” by B. S. Grewal, published by Khanna Publishers in 2018 (44th edition)
- “Higher Engineering Mathematics” by B. V. Ramana, published by Tata McGraw Hill Education in 2007″

**REFERENCE BOOKS:**

- “Advanced Engineering Mathematics” by Erwin Kreyszig, published by Wiley-India in 2015 (10th edition)
- “Advanced Engineering Mathematics with MATLAB” by Dean G. Duffy, published by CRC Press in 2010 (3rd edition)
- “Advanced Engineering Mathematics” by Peter O’Neil, published by Cengage in 2011 (7th edition)
- “Engineering Mathematics” by Samantha Pal and S C Bhunia, published by Oxford University Press in 2015″

**Course Outcomes:**

- Understand the physical significance of various operators such as gradient, curl, and divergence (at a level 5 of mastery)
- Calculate the work done against a field, circulation, and flux using vector calculus (at a level 5 of mastery)
- Use Laplace transforms to solve differential equations (at a level 3 of mastery)
- Find or compute the Fourier series of periodic signals (at a level 3 of mastery)
- Apply integral expressions for the forward and inverse Fourier transform to a variety of non-periodic waveforms (at a level 3 of mastery)
- Identify solution methods for partial differential equations that model physical phenomena (at a level 3 of mastery)”

HAI