Aligarh Muslim University (AMU) 2009 B.Sc Physics Mathematical Methods - Question Paper
I
4292
2008-2009
B.Sc.(HONS.) (PART-m) EXAMINATION (PHYSICS)
MATHEMATICAL METHODS (PH-307)
Maximum Marks: 40 Duration : Three Hours.
Answer all questions.
Marks are indicated against each part.
1.(a) Define complex integral. If f (z) is an analytic function in a simply connected domain D, then prove that
< i(z) dz = 2'JUf(Zo)
I z'zo
where z 0 is some point in the interior region bounded by curve C in domain D. 04
(b) Develop the Taylor series expansion of n (1 + z) for \z \ < 1. 03
2.(a) State and prove Divergence theorem of Gauss . 04 (b) Prove that line integral
J F(r) dr = Ft d* + F2 + F3 <*z ) c c _
is independent of path if and only of F = [ Fi, F2, F3 ] is a gradient of some
function f in D i.e. F = grad f. 03
OR
2 (a) Show that the scale factors h j corresponding to curvilinear coordinates q j for orthogonal system are given by the relation
where x k are the Cartesian coordinates. 05
(b) Calculate the Spherical polar coordinates scale factors h T, h e and h 02
3.(a) Define Gamma function in integral form and show that T(n+l) = n!. 03 (b) Establish the following differential formula for Hermite polynomials: 03
4.(a) For Legendre polynomial P n (x) prove that: 03
1
f Pn(x)Pm(*) dx = 0 for nm
(b) Show that n P n (x) = ( 2 n -1) x P n-i (x) - (n -1) P n _2 (x). 04
OR
4(a) Find the Associated Legendre equation from Legendre differential equation. 03
(b) Find the value of Spherical harmonics Y io . 04
5.(a) A triangular wave is represented by the function f (x) as
x if 0 <x < 7i
f (x) = <
-x if -n < x < 0.
Represent f (x) by a Fourier series. 03
(b) Consider a thin circular plate whose faces are impervious to heat flow and whose circular edge is kept at zero temperature. At t = 0, the initial temperature of the plate is a function of the distance from the centre of the plate. Find the expression for the subsequent temperature. 04
OR
5(a) Represent
'1 0 < x < XA
I 0 Vi < x < 1.
in a Fourier series in series and cosines. 03
(b) A bar of length and uniform cross section whose surface is impervious to heat flow has an initial temperature F (x). Its ends are kept at the constant temperature zero. Determine the subsequent temperature of the bar as time t increases. 04
6.(a) Obtain the solution of the non-homogeneous Fredholm equation of second kind
by separable Kernel method. 03
(b) Solve the equation
<p(x) = 1+ X | (J- 3xt) (pit) dt
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Attachment: |
| Earning: Approval pending. |
