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APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY

FIRST SEMESTER M.TECH DEGREE EXAMINATION, DECEMBER 201 7
Branch: Mechanical

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 Duration: 3 hours
PARTA

1 r
1. ದ್‌್‌ yj + 216, show that grad r = and grad( [இ] ಕನಡ 5 marks)

Evaluate fc (3x? — ‏برق‎ 2)dx + (4y — 6xy)dy where 6 is the boundary of the region by
y =andy = > 2 (4 marks)

2. a. Find div F and curl F where F = grad (x 3 + y 3 + 2 3 + 3xyz) ( 5 marks)

ன

A covariant tensor has components 21 — 2, ×2۷ andyz in rectangular
coordinates. Find its covariant components in spherical coordinates. (4 marks)

3. a. Find the components of the first and second fundamental tensors in cylindrical

coordinates. (4 marks )

If (ds) 2ಎ (dr) 2+ (427 + 72517126 (4क)2, find the values of [22, Wand ۰

22
(5 marks)
PART B
4. a. Show that the integral equation
1

y= —x 2 + y(t). t(t — x) dt
2 is equivalent to the differential equation y" (x) + xy(x)
= 1;y(0) = y (0) = 0
(4 marks )