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 �⃗� = 𝑥𝑦𝑧(𝑦𝑧𝑖̂ + 𝑥𝑧𝑗̂ + 𝑥𝑦�̂�) is

conservative. Hence find the scalar potential.

Q2. (a) In what direction from the point (1, 1, −1) is the directional derivative of

𝑓 = 𝑥2 − 2𝑦2 + 4𝑧2 a maximum? Also find the value of this maximum directional derivative.

(b) What is the greatest rate of increase of the temperature 𝑇(𝑥, 𝑦, 𝑧) = 𝑥𝑦𝑧2 at the point

(1, 0, 3).

Q3. Prove that ∇. {𝑟∇ (1

𝑟3)} =

3

𝑟4. . Further, write the physical meaning of Gradient, Divergence

and Curl.

Q4. Find the angle between the surfaces 𝑥 log 𝑧 = 𝑦2 − 1 and 𝑥2𝑦 = 2 − 𝑧 at the point (1, 1, 1).

Q5. (a)Show that ∫ (𝑦𝑧 − 1)𝑑𝑥𝑐

+ (𝑧 + 𝑥𝑧 + 𝑧2)𝑑𝑦 + (𝑦 + 𝑥𝑦 + 2𝑦𝑧)𝑑𝑧 is independent of

path of integration from (1, 2, 2) to (2, 3, 4). Evaluate the integral.

(b) Evaluate ∫ 𝑥𝑦3𝑑𝑠𝑐

, where 𝐶 is the segment of the line 𝑦 = 2𝑥 in the 𝑥𝑦 plane from

(−1,−2) to (1, 2) and 𝑠 is the arc length.

Q6. If 𝐹 = 𝑦𝑖̂ − 𝑥𝑗̂ evaluate ∫ 𝐹 . 𝑑𝑟 𝐶

from (0, 0) to (1, 1) along the following paths:

(a) The parabola 𝑦 = 𝑥2

(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 ∫ 𝐴 . �̂� 𝑑𝑆𝑆

where 𝐴 = 𝑦𝑧 𝑖̂ + 𝑧𝑥𝑗̂ + 𝑥𝑦�̂� and 𝑆 is the part of the sphere

𝑥2 + 𝑦2 + 𝑧2 = 9 which lies in the first octant.

Q8. Verify divergence theorem for 𝐹 = (𝑥2 − 𝑦𝑧)𝑖̂ + (𝑦2 − 𝑧𝑥)𝑗̂ + (𝑧2 − 𝑥𝑦)�̂� taken over the

rectangular parallelepiped.

Q9. Verify Green’s Theorem in the plane for the integral

∮ (3𝑥2 − 8𝑦2)𝑑𝑥 + 4(4𝑦 − 6𝑥𝑦)𝑑𝑦𝐶

where 𝐶 is the boundary of the region bounded by 𝑦 = √𝑥 and 𝑦 = 𝑥2.

Q10. Evaluate by Green’s Theorem

∮ (cos 𝑥 sin 𝑦 − 𝑥𝑦)𝑑𝑥 + (sin 𝑥 cos 𝑦)𝑑𝑦𝐶

where C is the circle 𝑥2 + 𝑦2 = 1.

Q11. Introducing 𝐴 = 𝑁𝑖̂ − 𝑀𝑗̂, show that the formula in Green’s theorem may be written as

∬ div𝐴 𝑅

𝑑𝑥𝑑𝑦 = ∮ 𝐴 . �̂�𝑑𝑠𝐶

where �̂� is the outward normal vector to C and 𝑠 is the arc length.