DEPARTMENT OF MATHEMATICS
College of Engineering Studies
University of Petroleum & Energy Studies, Dehradun, Uttarakhand
Programe: B.Tech. (All Branches)
Subject: Mathematics II Subject Code: MATH 102
Assignment: II Semester: II
Topic: Vector Calculus
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Q1. Show that the vector field defined by the vector function ๏ฟฝโ๏ฟฝ = ๐ฅ๐ฆ๐ง(๐ฆ๐ง๐ฬ + ๐ฅ๐ง๐ฬ + ๐ฅ๐ฆ๏ฟฝฬ๏ฟฝ) is
conservative. Hence find the scalar potential.
Q2. (a) In what direction from the point (1, 1, โ1) is the directional derivative of
๐ = ๐ฅ2 โ 2๐ฆ2 + 4๐ง2 a maximum? Also find the value of this maximum directional derivative.
(b) What is the greatest rate of increase of the temperature ๐(๐ฅ, ๐ฆ, ๐ง) = ๐ฅ๐ฆ๐ง2 at the point
(1, 0, 3).
Q3. Prove that โ. {๐โ (1
๐3)} =
3
๐4. . Further, write the physical meaning of Gradient, Divergence
and Curl.
Q4. Find the angle between the surfaces ๐ฅ log ๐ง = ๐ฆ2 โ 1 and ๐ฅ2๐ฆ = 2 โ ๐ง at the point (1, 1, 1).
Q5. (a)Show that โซ (๐ฆ๐ง โ 1)๐๐ฅ๐
+ (๐ง + ๐ฅ๐ง + ๐ง2)๐๐ฆ + (๐ฆ + ๐ฅ๐ฆ + 2๐ฆ๐ง)๐๐ง is independent of
path of integration from (1, 2, 2) to (2, 3, 4). Evaluate the integral.
(b) Evaluate โซ ๐ฅ๐ฆ3๐๐ ๐
, where ๐ถ is the segment of the line ๐ฆ = 2๐ฅ in the ๐ฅ๐ฆ plane from
(โ1,โ2) to (1, 2) and ๐ is the arc length.
Q6. If ๐น = ๐ฆ๐ฬ โ ๐ฅ๐ฬ evaluate โซ ๐น . ๐๐ ๐ถ
from (0, 0) to (1, 1) along the following paths:
(a) The parabola ๐ฆ = ๐ฅ2
(b) The straight lines from (0, 0) to (1, 0) and then to (1, 1).
(c) The straight line joining (0, 0) and (1, 1)
Q7. Evaluate โซ ๐ด . ๏ฟฝฬ๏ฟฝ ๐๐๐
where ๐ด = ๐ฆ๐ง ๐ฬ + ๐ง๐ฅ๐ฬ + ๐ฅ๐ฆ๏ฟฝฬ๏ฟฝ and ๐ is the part of the sphere
๐ฅ2 + ๐ฆ2 + ๐ง2 = 9 which lies in the first octant.
Q8. Verify divergence theorem for ๐น = (๐ฅ2 โ ๐ฆ๐ง)๐ฬ + (๐ฆ2 โ ๐ง๐ฅ)๐ฬ + (๐ง2 โ ๐ฅ๐ฆ)๏ฟฝฬ๏ฟฝ taken over the
rectangular parallelepiped.
Q9. Verify Greenโs Theorem in the plane for the integral
โฎ (3๐ฅ2 โ 8๐ฆ2)๐๐ฅ + 4(4๐ฆ โ 6๐ฅ๐ฆ)๐๐ฆ๐ถ
where ๐ถ is the boundary of the region bounded by ๐ฆ = โ๐ฅ and ๐ฆ = ๐ฅ2.
Q10. Evaluate by Greenโs Theorem
โฎ (cos ๐ฅ sin ๐ฆ โ ๐ฅ๐ฆ)๐๐ฅ + (sin ๐ฅ cos ๐ฆ)๐๐ฆ๐ถ
where C is the circle ๐ฅ2 + ๐ฆ2 = 1.
Q11. Introducing ๐ด = ๐๐ฬ โ ๐๐ฬ, show that the formula in Greenโs theorem may be written as
โฌ div๐ด ๐
๐๐ฅ๐๐ฆ = โฎ ๐ด . ๏ฟฝฬ๏ฟฝ๐๐ ๐ถ
where ๏ฟฝฬ๏ฟฝ is the outward normal vector to C and ๐ is the arc length.