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Test: Divergence Theorem - Civil Engineering (CE) MCQ


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15 Questions MCQ Test Engineering Mathematics - Test: Divergence Theorem

Test: Divergence Theorem for Civil Engineering (CE) 2024 is part of Engineering Mathematics preparation. The Test: Divergence Theorem questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: Divergence Theorem MCQs are made for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Divergence Theorem below.
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Test: Divergence Theorem - Question 1

Given a function ϕ = 1/2(x+ y+ z2) in three-dimensional Cartesian space, the value of the surface integral ∯S n̂ . ∇ϕ dS where S is the surface of a sphere of unit radius and n̂ is the outward unit normal vector on S, is

Detailed Solution for Test: Divergence Theorem - Question 1

Gauss divergence theorem:

It states that the surface integral of a vector field over a closed surface is equal to the volume integral of the divergence of that vector field over the volume enclosed by the closed surface.

∯S A.dS =  ∫∫∫V (∇.A) dV

Calculation:

Given:

S = surface of sphere , V = volume of sphere = 4/3πr3

r = radius of sphere = 1

Using Gauss theorem 

∯S n̂ . ∇ϕ dS  = 4π 

Test: Divergence Theorem - Question 2

The value of  where S is the surface of unit sphere x2 + y2 + z2 = 1 is

Detailed Solution for Test: Divergence Theorem - Question 2

According to the divergence theorem,

In expanded form,

Calculation:

Given Fx = yz, Fy = zx, Fz = xy;

Substituting in the theorem,

⇒ 

⇒ 

⇒ I = 0

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Test: Divergence Theorem - Question 3

Detailed Solution for Test: Divergence Theorem - Question 3

Using Gauss-Divergence theorem

F̅ = ϕ∇ψ – ψ∇ϕ

∇⋅F̅  = ∇ ⋅ [ϕ∇ψ – ψ∇ϕ]

Now,

∇ ⋅ (ϕ∇ψ) = ∇ϕ ⋅ ∇ψ + ϕ∇ ⋅ ∇ψ

= ∇ϕ ⋅ ∇ψ + ϕ∇2ψ

∇ ⋅ (ψ∇ϕ) = ∇ψ ⋅ ∇ϕ + ψ∇ ⋅ ∇ϕ

= ∇ψ ⋅ ∇ϕ + ψ∇2ϕ

∴ ∇ ⋅ F̅ = ∇ϕ ⋅ ∇ψ + ϕ∇2ψ - ∇ψ ⋅ ∇ϕ – ψ∇2ψ

∇ ⋅ F̅ = ϕ∇2ψ – ψ∇2ϕ

∴ 

Test: Divergence Theorem - Question 4

Consider a closed surface S surrounding volume V. If is the position vector of a point inside S, with the unit normal  on S, the value of the integral  is

Detailed Solution for Test: Divergence Theorem - Question 4

From Divergence theorem, we have

The position vector 

Here, 

Thus, 

So, 

= 15V

*Answer can only contain numeric values
Test: Divergence Theorem - Question 5

Find the value of  ds where  and S is the surface of sphere x+ y+ z2 = 16 ____


Detailed Solution for Test: Divergence Theorem - Question 5

From the Gauss Divergence theorem we have

∴ 

Given radius of sphere r = √16 = 4

= 804.248

Test: Divergence Theorem - Question 6

The divergence theorem for a surface consisting of a sphere is computed in which coordinate system?

Detailed Solution for Test: Divergence Theorem - Question 6

Seeing the surface as sphere, we would immediately choose spherical system, but it is wrong. The divergence operation is performed in that coordinate system in which the function belongs to. It is independent of the surface region.

Test: Divergence Theorem - Question 7

The divergence theorem value for the function x2 + y2 + z2 at a distance of one unit from the origin is

Detailed Solution for Test: Divergence Theorem - Question 7

Div (F) = 2x + 2y + 2z.
The triple integral of the divergence of the function is ∫∫∫(2x + 2y + 2z)dx dy dz, where x = 0->1, y = 0->1 and z = 0->1. On integrating,
we get 3 units.

Test: Divergence Theorem - Question 8

For a function given by F = 4x i + 7y j +z k, the divergence theorem evaluates to which of the values given, if the surface considered is a cone of radius 1/2π m and height 4π2 m.

Detailed Solution for Test: Divergence Theorem - Question 8

Div (F) = 4 + 7 + 1 = 12. The divergence theorem gives ∫∫∫(12).dV, where dV is the volume of the cone πr3h/3, where r = 1/2π m and h = 4π2 m. On substituting the radius and height in the triple integral,
we get 2 units.

Test: Divergence Theorem - Question 9

Divergence theorem is based on

Detailed Solution for Test: Divergence Theorem - Question 9

The divergence theorem relates surface integral and volume integral. Div(D) = ρv, which is Gauss’s law.

Test: Divergence Theorem - Question 10

Find the divergence theorem value for the function given by (ez, sin x, y2)

Detailed Solution for Test: Divergence Theorem - Question 10

Since the divergence of the function is zero, the triple integral leads to zero. The Gauss theorem gives zero value.

*Answer can only contain numeric values
Test: Divergence Theorem - Question 11

The value of the surface integral  over the surface S of the sphere x2 + y2 + z2 = 9, where n is the unit outward normal to the surface element dS, is _______.


Detailed Solution for Test: Divergence Theorem - Question 11

Gauss divergence theorem:

  • It states that the surface integral of the normal component of a vector function  taken over a closed surface ‘S’ is equal to the volume integral of the divergence of that vector function  taken over a volume enclosed by the closed surface ‘S’.
  • Mathematically, it can be written as:

Calculation:

Given:
The surface of the given sphere is S = x2 + y2 + z2 = 9, it is a closed surface,

= = 216π = 678.580

Test: Divergence Theorem - Question 12

The value of ∯ (4xî - 2y2j + z2k).n̂ds where S is bounded by x2 + y2 = 4, Z = 0 and Z  = 3 is

Detailed Solution for Test: Divergence Theorem - Question 12

From Gauss – Divergence theorem,

 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

Test: Divergence Theorem - Question 13

The divergence of the vector field 

Detailed Solution for Test: Divergence Theorem - Question 13

Concept:

The divergence of any vector field 
is defined as:

The nabla operator is defined as:

Calculation:

Given:

Vector 

Divergence of u will be

Test: Divergence Theorem - Question 14

Identify the nature of the field, if the divergence is zero and curl is also zero.

Detailed Solution for Test: Divergence Theorem - Question 14

Since the vector field does not diverge (moves in a straight path), the divergence is zero. Also, the path does not possess any curls, so the field is irrotational.

Test: Divergence Theorem - Question 15

Find whether the vector is solenoidal, E = yz i + xz j + xy k

Detailed Solution for Test: Divergence Theorem - Question 15

Div(E) = Dx(yz) + Dy(xz) + Dz(xy) = 0. The divergence is zero, thus vector is divergentless or solenoidal.

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