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Sma2109:Applied Mathematics Question Paper

Sma2109:Applied Mathematics

Course:Bachelor Of Computer Science

Institution: Meru University Of Science And Technology question papers

Exam Year:2013

QUESTION ONE (30 MARKS)
(a) Write out all the tensor in ?? = ?????????????? taking ?? = 3. (3 Marks) (b) (i) Define a Covariant tensor of rank one. (ii) Show that if p qA and r B are tensors, r p q BA is also a tensor. (5 Marks)
(c) Find the inverse Laplace transform of the function ? ? 1 1 )( 2 ? ? ss sf (6 Marks)
(d) (i) Prove that Bessel’s function of order zero is given by ? ? ... 642422 1 2 22 6 22 4 2 2 0 ? ?? ? ? ??? x xx xJ (4 Marks)
(ii) State the Recurrence formula for Legendre Polynomials and use it to find ??2 ?? . (5 Marks) (e) (i) Define an integral equation. (2 Marks) (ii) Solve the integral equation. ? ? ? ?ds xeex x sx ? ? ?? 0 (5 Marks)
2
QUESTION TWO (20 MARKS)
(a) A covariant tensor has components ????, 2?? - ??2,???? in rectangular coordinates. Find its covariant components in spherical coordinates. (15 Marks) (b) Determine the metric tensor in matrix form in cylindrical coordinates. (5 Marks)
QUESTION THREE (20 MARKS)
(a) Giving an example for each, write down (i) The Fredholm integral equation of the first kind. (2 Marks) (ii) The Volteria integral equation of the first kind. (2 Marks) (b) Form the integral equation corresponding to ??'' + ????' + ?? = 0 ?? ?? = 2, ??' 0 = 1 (8 Marks)
(c) Solve the initial value problem using Laplace transform ??2?? ????2 - 2 ???? ???? - 8?? = 0 ?? 0 = 0, ??' 0 = 6 (8 Marks)
QUESTION FOUR (20 MARKS)
(a) Solve the integral equation ? ? ? ?ds ssxx ??? 2 0 21 ?? ? (8 Marks) (b) State four problems which give rise to integral equations. (4 Marks) (c) Write the normalized form of the differential equation and state ? ? ? ? xpandxp 21 . ? ? ? ? ? ? 0 1222 2 2 22 ? ????? y x dx dy x dx yd xx (3 Marks) (d) Define the Gamma function and show that G1 = 1 (5 Marks)

Sma2109:Applied Mathematics Question Paper

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