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Students those who are studying JNTUK R20 Mechanical Branch, Can Download Unit wise R20 2-1 Vector Calculus, Fourier Transforms and PDE(M-III) Material/Notes PDFs below.

Course Objectives : At the end of the course, the student will be able to

• Interpret the physical meaning of different operators such as gradient, curl and divergence (L5)
• Estimate the work done against a field, circulation and flux using vector calculus (L5)
• Apply the Laplace transform for solving differential equations (L3).
• Find or compute the Fourier series of periodic signals (L3)
• Know and be able to apply integral expressions for the forwards and inverse Fourier transform to a range of non-periodic waveforms (L3)
• Identify solution methods for partial differential equations that model physical processes (L3)

UNIT-1

Vector calculus:

Vector Differentiation: Gradient — Directional derivative — Divergence — Curl — Scalar Potential.

Vector Integration: Line integral — Work done — Area — Surface and volume integrals — Vector integral theorems: Greens, Stokes and Gauss Divergence theorems (without proof).

UNIT-2

Laplace Transforms: Laplace transforms of standard functions — Shifting theorems — Transforms of derivatives and integrals — Unit step function — Dirac’s delta function — Inverse Laplace transforms — Convolution theorem (without proof). Applications: Solving ordinary differential equations (initial value problems) using Laplace transforms.

UNIT-3

Fourier series and Fourier Transforms: Fourier Series: Introduction — Periodic functions — Fourier series of periodic function — Dirichlet’s conditions — Even and odd functions — Change of interval — Half-range sine and cosine series. Fourier Transforms: Fourier integral theorem (without proof) — Fourier sine and cosine integrals — Sine and cosine transforms — Properties — inverse transforms — Finite Fourier transforms.

UNIT-4

PDE of first order: Formation of partial differential equations by elimination of arbitrary constants and arbitrary functions — Solutions of first order linear (Lagrange) equation and nonlinear (standard types) equations.

UNIT-5

Second order PDE and Applications: (10 hrs) Second order PDE: Solutions of linear partial differential equations with constant coefficients — RHS term of the type eax + by ,sin(ax + by), cos(ax + by), xm y n

Applications of PDE: Method of separation of Variables — Solution of One dimensional Wave, Heat and two-dimensional Laplace equation.

Text Books:

1. B.S. Grewal, Higher Engineering Mathematics, 43rd Edition, Khanna Publishers.

2. B. V. Ramana, Higher Engineering Mathematics, 2007 Edition, Tata Mc. Graw Hill Education.

Reference Books:

1. Erwin Kreyszig, Advanced Engineering Mathematics, 10th Edition, Wiley-India.

2. Dean. G. Duffy, Advanced Engineering Mathematics with MATLAB, 3rd Edition, CRC Press.

3. Peter O’ Neil, Advanced Engineering Mathematics, Cengage.

4. Srimantha Pal, S C Bhunia, Engineering Mathematics, Oxford University Press.