Divergence Of a Vector Field ( div A) - Definition, Significance & Solved Problems.

Mathematically divergence of a vector A is defined as:

Where the surface S is a closed surface that completely surrounds a small volume Δ v and where ds points outward from the closed surface.


- A divergence is applied to a vector as a function of position, yielding a scalar. The divergence actually measures how much the vector function is spreading out.


- Divergence of a vector field A is a measure of how much a vector field converges to or diverges from a given point. In simple terms it is a measure of the outgoingness of a vector field.


- Divergence of a vector field is positive if the vector diverges or spread out from a given point. If you are at a location from which the vector field tends to point away in all directions, you will definitely have a positive divergence. It means divergence is positive at a source point in the field.


- Divergence of a vector field is negative if the vector field converges at that point. If the field points inward toward a point, the divergence in and near that point is negative. It means divergence is negative at sink point in the field.


- Hence a nonzero divergence at some point means there must be a source or sink at that position.


- If just as much of the vector field points in as out, the divergence will be approximately zero.


- A vector field with constant zero divergence is called solenoidal or divergenceless or incompressible (∇ . A = 0). In such cases no net flow can occur across any closed surface.


- Divergence of a vector field results in a scalar field that represents the sources of the vector field.


- Divergence of a vector field A in Cartesian coordinate system is given as:

 

 



- Divergence of a vector field A in Cylindrical coordinate system is given as:

- Divergence of a vector field A in Spherical coordinate system is given as:

DIVERGENCE THEOREM:

- It states that the net outward flux of a vector field A through a closed surface S is equal to the volume integral of the divergence of the field A inside the surface.

- It also states that the sum of all sources minus the sum of all sinks gives the net flow out of a region.



SOLVED EXAMPLES / NUMERICALS:

Q.10 Determine the divergence of the following vector fields and evaluate them at the specified points:

a) A = yz a_x + 4xy a_y + y a_z at point (1, -2, 3)
b) A = ρzsinφ a_ρ + 3ρz² cosφ a_φ at (5, π / 2, 1)
c) A = 2r cosθ cosφ a_r + r^1/2 a_φ   at (1, π / 6, π / 3)    SOLUTION/ANSWER


Q.11 Determine the flux of D = ρ² cos²φ a_ρ + zsin φ a_φ over the closed surface of the cylinder, 0 < z < 1, ρ =4. Verify the divergence theorem for this case.       SOLUTION/ANSWER


ALSO READ:

- Line , Surface and Volume Intergral.

- Del Operator - Definition and Significance.

- Gradient Of a Scalar (∇ V).

- Numericals / Solved Examples - Gradient Of a Scalar.

- Divergence Of a Vector ( ∇ . A ).

- Numericals / Solved Examples - Divergence Of a Vector.

- Curl Of a Vector ( ∇ x A).

- Laplacian Of a Scalar ( ∇² V).


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