Test: Divergence Theorem - Civil Engineering (CE) MCQ

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 

According to the divergence theorem,

In expanded form,

Calculation:

Given F_x = yz, F_y = zx, F_z = xy;

Substituting in the theorem,

⇒ 

⇒ 

⇒ I = 0

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

∴ 

From Divergence theorem, we have

The position vector 

Here, , 

Thus, 

So, 

= 15V

From the Gauss Divergence theorem we have

∴ 

Given radius of sphere r = √16 = 4

= 804.248

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.

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.

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.

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

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

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

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

Concept:

The divergence of any vector field 
is defined as:

The nabla operator is defined as:

Calculation:

Given:

Vector 

Divergence of u will be

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.

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