Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > The value of(4xi - 2y2j + z2k).ndswhere S is ...
Start Learning for Free
The value of ∯ (4xî - 2y2j + z2k).n̂ds where S is bounded by x2 + y2 = 4, Z = 0 and Z = 3 is
- a)84 π
- b)80 π
- c)∰ (4 - 4r sin θ + 2z) rdr dθ dz where x = r cos θ, y = r sin θ
- d)∰ (4 - 4r sin θ + 2z) r2 dr dθ dz where x = r cos θ, y = r sin θ
Correct answer is option 'A'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z =...
From Gauss – Divergence theorem,
∯
n̂ds = ∭ (∇ ⋅ F) dV
n̂ds = ∭ (∇ ⋅ F) dV ∯ (4x̂i - 2y2j + z2k). n̂ds = ∭ ∇ ⋅ (4x̂i - 2y2j + z2k) dV
= ∭ (4 - 4y + 2z) dx dy dz
By change of variables,
x = r cos θ
y = r sin θ
z = z
dx dy dz = r dr dθ dz
Limits
z = 0 → z = 3
r = 0 → r = 2
θ = 0 → θ = 2π
= 16π (3) + 4π (9)
= 84 π
Free Test
FREE
| Start Free Test |
Community Answer
The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z =...
To find the value of the given vector field (4xi - 2y^2j + z^2k) · ds, where S is bounded by x^2 + y^2 = 4, z = 0, and z = 3, we can use the surface integral formula:
∫∫(F · n) ds
where F is the vector field, n is the unit normal vector to the surface, and ds is the surface area element.
Since the given surface is a portion of a cylinder, we can use cylindrical coordinates to parametrize the surface. Let's use the following parameterization for S:
r(θ, z) = (2cosθ)i + (2sinθ)j + zk
where 0 ≤ θ ≤ 2π and 0 ≤ z ≤ 3.
Now, let's find the partial derivatives of r with respect to θ and z:
∂r/∂θ = (-2sinθ)i + (2cosθ)j + 0k
∂r/∂z = 0i + 0j + k
Next, let's find the cross product of these partial derivatives to find the unit normal vector:
n = (∂r/∂θ) x (∂r/∂z)
= (-2sinθ)i + (2cosθ)j + 0k
Now, let's evaluate the dot product (F · n):
(F · n) = (4xi - 2y^2j + z^2k) · (-2sinθ)i + (2cosθ)j + 0k
= -8xsinθ + 4y^2cosθ + z^2(0)
= -8(2cosθ)sinθ + 4(2sinθ)^2cosθ + z^2(0)
= -16cosθsinθ + 16sin^2θcosθ
= 16sin^2θcosθ - 16cosθsinθ
= 0
Since the dot product (F · n) is 0, the value of the given vector field integrated over the surface S is 0.
Therefore, the correct answer is 0.
∫∫(F · n) ds
where F is the vector field, n is the unit normal vector to the surface, and ds is the surface area element.
Since the given surface is a portion of a cylinder, we can use cylindrical coordinates to parametrize the surface. Let's use the following parameterization for S:
r(θ, z) = (2cosθ)i + (2sinθ)j + zk
where 0 ≤ θ ≤ 2π and 0 ≤ z ≤ 3.
Now, let's find the partial derivatives of r with respect to θ and z:
∂r/∂θ = (-2sinθ)i + (2cosθ)j + 0k
∂r/∂z = 0i + 0j + k
Next, let's find the cross product of these partial derivatives to find the unit normal vector:
n = (∂r/∂θ) x (∂r/∂z)
= (-2sinθ)i + (2cosθ)j + 0k
Now, let's evaluate the dot product (F · n):
(F · n) = (4xi - 2y^2j + z^2k) · (-2sinθ)i + (2cosθ)j + 0k
= -8xsinθ + 4y^2cosθ + z^2(0)
= -8(2cosθ)sinθ + 4(2sinθ)^2cosθ + z^2(0)
= -16cosθsinθ + 16sin^2θcosθ
= 16sin^2θcosθ - 16cosθsinθ
= 0
Since the dot product (F · n) is 0, the value of the given vector field integrated over the surface S is 0.
Therefore, the correct answer is 0.
Attention Civil Engineering (CE) Students!
To make sure you are not studying endlessly, EduRev has designed Civil Engineering (CE) study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Civil Engineering (CE).
|
Explore Courses for Civil Engineering (CE) exam
|
|
Similar Civil Engineering (CE) Doubts
Top Courses for Civil Engineering (CE)View all
The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer?
Question Description
The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer?.
The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer?.
Solutions for The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for Civil Engineering (CE).
Download more important topics, notes, lectures and mock test series for Civil Engineering (CE) Exam by signing up for free.
Here you can find the meaning of The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer?, a detailed solution for The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The value of(4xi - 2y2j + z2k).ndswhere S is bounded by x2+ y2= 4, Z = 0 and Z = 3 isa)84 πb)80 πc)(4 - 4r sin θ + 2z) rdr dθdzwherex = r cosθ, y = r sinθd)(4 - 4r sinθ + 2z) r2dr dθ dz wherex = r cosθ, y = r sinθCorrect answer is option 'A'. Can you explain this answer? tests, examples and also practice Civil Engineering (CE) tests.
|
|
Explore Courses for Civil Engineering (CE) exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup