No. of pages: 2

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY

FIRST SEMESTER M TECH DEGREE EXAMINATION, DECEMBER
201 8

Branch: Mechanical Engineering
Stream: Machine Design

Course Code & Name: 01 MA601 1 Special Functions, Partial Differential Equations and
Tensors

Answer any two full questions from each part
Limit answers to the required points

Max. Marks: 60

PartA

I (೩). Evaluate fs F.n dS where F = 4xi — 292] + 721 and S is the surface bounding the
region x 7+ ४2 = 4, 2 = 0 2142 = 3. (3 marks)

(0). State Divergence theorem (3 marks) (0). Define: Gradient, Divergence and Curl (3

marks)

2 (a). Using Green's theorem evaluate y 7 dx + x *dy where C is the square with vertices
(0,0), (1.0), 6, 1) and (0,1) oriented counter clockwise. (6 marks)
(b) Define (i) contravariant tensor oforder ]

(ii) covariant tensor of order |
(iii) mixed tensor of order 2 (3 marks)

3 (a). Find the components of the first and second fundamental tensors in spherical

coordinates (6 marks)

(b). Show that the velocity of a fluid at any point is a contravariant tensor ofrank I (3 marks)
Part B

4 (a). Transform the IVP y" + xy' + y = 0, y(0) = 1, and y'(O) = 0 into an integral

equation. (5 marks)