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Physics Exam: Mathematical Physics

This document appears to be an exam for a third semester physics course covering mathematical physics. It contains 22 multiple choice and written response questions assessing students' knowledge of topics like: 1) defining analytic functions, 2) separating real and imaginary parts of complex expressions, 3) using theorems like Stokes' theorem and Gauss' integral formula, 4) matrix operations and properties, 5) Fourier series and integrals, 6) vector calculus concepts like divergence and curl, and 7) numerical integration techniques like Simpson's rule. The exam is worth a total of 100 marks and is divided into three parts with varying point values for the different question types.

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selvam1981
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118 views2 pages

Physics Exam: Mathematical Physics

This document appears to be an exam for a third semester physics course covering mathematical physics. It contains 22 multiple choice and written response questions assessing students' knowledge of topics like: 1) defining analytic functions, 2) separating real and imaginary parts of complex expressions, 3) using theorems like Stokes' theorem and Gauss' integral formula, 4) matrix operations and properties, 5) Fourier series and integrals, 6) vector calculus concepts like divergence and curl, and 7) numerical integration techniques like Simpson's rule. The exam is worth a total of 100 marks and is divided into three parts with varying point values for the different question types.

Uploaded by

selvam1981
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.DEGREE EXAMINATION –PHYSICS


THIRD SEMESTER – APRIL 2018
PH 3506– MATHEMATICAL PHYSICS

Date: 05-05-2018 Dept. No. Max. : 100 Marks


Time: 01:00-04:00
Part –A
Answer all questions (10× 𝟐 = 𝟐𝟎marks )

1. Define an analytic function

2. Separate the following into real and imaginary part of sin ( x  iy )

3. Find the unit normal to the surface x2+y2=z at point (1,2,5)

4. State Stoke’s theorem.

5. Define Euler coefficients for even half range expansion

∞ cosω 𝑑𝜔
6. Using Laplace integral, evaluate ∫0
1+𝜔2

7. What is a triangular matrix? Give an example

1 1 1
8. Determine the rank of a matrix [1 −1 −1]
3 1 1

9. Express Gauss’ integral formula and give its importance.

10. Write down the difference operator for f(x) by ‘h’.

Part- B

Answer any four questions. (4× 𝟕. 𝟓 = 𝟑𝟎 marks)

11. (a) Show that|𝑍 − 𝑖|2 =1 describes a circle centered at the (0,i) with radius 1.

(b) Simplify (1+i) (2+i) and locate it in the complex plane.

12. Using Green’s theorem, evaluate ∫𝑐 (𝑥 2 𝑦 𝑑𝑥 + 𝑥 2 𝑑𝑦 ) where C is boundary described counter –

clock wise of the triangle with vertices (0,0) (1,0), (1,1).

13. Obtain a Fourier expression for f(x) = x for -𝜋 < 𝑥 < 𝜋.

1 2 0
14. Verify Cayley – Hamilton theorem for the matrix (2 −1 0) and find its inverse.
0 0 1

1
15. Using Lagrange’s interpolation formula, find the value of Y when X=10 from the following data.

X 5 6 9 11

Y 12 13 14 16

𝑑𝑧
16. Use Cauchy’s integral theorem to evaluate the integral ∮𝐶 where 𝐶: |𝑧 + 𝑖| = 1 in the counter
𝑧 2 +1

clockwise direction.

Part –C

Answer any four questions. (4× 𝟏𝟐. 𝟓 = 𝟓𝟎 marks)

17. Establish that the real and complex part of an analytic function satisfies the Laplace equation.

18. (a) Prove that ∇. ∇ × 𝐹 = 0, where F is a three dimensional vector in Cartesian coordinates.

(b) Using Gauss –divergence theorem, evaluate ∬𝑆 ( 𝑥 3 𝑑𝑦𝑑𝑧 + 𝑦 3 𝑑𝑧𝑑𝑥 + 𝑧 3 𝑑𝑥𝑑𝑦) where S is the

surface of the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 = 4.

19. write down the functional form of a square wave of period 2𝜋 and obtain its Fourier series.

2 0 −2
20. Determine the eigen values of A =[ 0 0 −2] and show that matrix A satisfies its own
−2 −2 1

characteristic equation.
+3 1 rd
21. Calculate the approximate value of ∫−3 𝑥 4 𝑑𝑥 by Simpon’s rule. Compare it with the exact value
3

and the value obtained by Trapezoidal.

22. (a) Find the directional derivate of g =(𝑥 2 + 𝑦 2 + 𝑧 2 )-1/2 at (4,2,-4) in the direction of (1,2,-2).

⃗ = 𝑦𝑧 𝑖̂ + 𝑧𝑥 𝑗̂ + 𝑥𝑦 𝑘̂ and f = xyz, find curl (f𝑈


(b) If 𝑈 ⃗)

**********

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