The document discusses the concept of divergence, which describes how a vector field diverges from sources and sinks. Specifically, the divergence of an electric field yields the charge distribution that produces it. The divergence of a velocity field also provides a measure of how much the velocity spreads out from a point. Some examples are given, such as the divergence of the position vector equalling 3, and the conditions for a vector field to be solenoidal (having zero divergence). The physical interpretation is that the divergence quantifies the net rate of flow of a fluid out of a small volume, and can be written as the derivative of the product of the fluid density and velocity.

1. NPTEL – Physics – Mathematical Physics - 1
Lecture 3
Divergence
The divergence of a vector field describes how the field diverges from the “sources” and “sinks” of the
field. Specifically in electromagnetics, divergence of electric field yields the charge distribution
that produces it. In a everyday example, the water flowing in a pipe with a constant velocity 𝑣 (across the
cross section of the pipe) and pointing along its axis has to have vector function associated with it where
𝑣 . 𝑑𝑠 where 𝑑𝑠 is the differential area equal to nˆ 𝑑𝑠 ( nˆ is the unit drawn normal) is the flux of water
(or the water collected in a bucket). A divergence of the velocity field also produces a measure of how
much
𝑣 spreads out from a point.
We have just seen scalar derivatives of a vector, such as , and vector derivatives of scalar, such as
𝑑𝑟
⃗𝑑𝑡
∇⃗⃗𝜑. Now we shall see vector derivatives of a vector, such
as
∇⃗⃗. 𝑣⃗
=
𝑑𝑣𝑥
+
𝑑𝑣𝑦
+
𝑑𝑣𝑧
𝑑𝑥 𝑑𝑦 𝑑𝑧
∇⃗⃗. 𝑣⃗ is a
scalar.
Examples
∇⃗⃗. 𝑟⃗ =
(𝑥̂
𝑑
𝑑𝑥
+ ŷ
𝑑 𝑑
+ ẑ ) . (𝑥 x̂ + 𝑦 ŷ + 𝑧 ẑ )
𝑑𝑦 𝑑𝑧
= 3
So the divergence of the position vector yields unity corresponding to each direction in space.
1. ⃗∇⃗. (𝑓𝑣⃗ ) =
𝑑
(𝑓𝑣 ) + (𝑓𝑣 ) + 𝑑
(𝑓𝑣 )
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𝑑𝑥 𝑑𝑦 𝑑𝑧
𝑥 𝑦 𝑧
𝑑
When 𝑓 is an arbitrary scalar. Thus
⃗∇⃗. (𝑓𝑣⃗) = (𝑣𝑥 + 𝑓 ) + (𝑦 term) + (𝑧 term)
𝑑𝑓
𝑑𝑥 𝑑𝑥
𝑑𝑣𝑥
= (∇⃗⃗𝑓). 𝑣⃗ + 𝑓(⃗∇⃗. 𝑣̂ )
Also for a constant surface 𝜑, 𝑑𝜑 = 0. Which yield ⃗∇⃗𝜑 being
perpendicular to 𝑑𝑙 .
The divergence arises in a wide variety of physical situations, such as
∇⃗⃗. 𝐵⃗⃗ = 0 where 𝐵⃗⃗ is the magnetic induction
⁄ 0 where 𝐸 is the electrostatic field and 𝜌 is the charge density.
∇⃗⃗. 𝐸⃗⃗ =

𝜀
⃗
⃗

2. NPTEL – Physics – Mathematical Physics - 1
Physical interpretation of divergence
Let us try to understand the physical meaning of the word ‘divergence’. It quantifies how much a vector
field diverges in space.
Let us consider a compressible fluid with velocity 𝑣⃗ (𝑥, 𝑦, 𝑧) and a density 𝜌(𝑥, 𝑦, 𝑧).
The volume of fluid flowing into the parallelepiped shown per unit time through the face EFGH in
𝑣𝑥 | 𝑥=0𝑑𝑦𝑑𝑧. Further, the volume of fluid coming out of the face ACDB is 𝜌𝑣𝑥 |𝑥 𝑑𝑦𝑑𝑧.
Assuming 𝑑𝑥 is small and the 𝑣𝑥 |𝑥 can be computed from 𝑉
𝑥 |𝑥=0 using Taylor’s series. Thus
𝜌𝑣𝑥 |𝑥=𝑑𝑥 𝑑𝑦𝑑𝑧 = [𝜌𝑣𝑥 + 𝑑𝑥
(𝜌𝑣𝑥 )𝑑𝑥]
𝑥=0
𝑑𝑦𝑑𝑧.
Thus the net rate of fluid flowing out is
𝑑
𝑑
(𝜌𝑣 )𝑑𝑥𝑑𝑦𝑑𝑧. The above can be written as
𝑑𝑥 𝑥
𝑙𝑡
∆𝑥 → 0
𝜌𝑣𝑥 (∆𝑥, 0, 0) − 𝜌𝑣𝑥
(0,0,0) ∆𝑥
= [𝜌𝑣]
𝑑
𝑑𝑥
Considering the other two directions, the rate of flow out is ⃗∇⃗. (𝜌𝑣⃗)𝑑𝑥𝑑𝑦𝑑𝑧.
A vector field 𝐴⃗ which does not diverge at all, is called solenoidal for which ⃗∇⃗. 𝐴
⃗=0. Consider an electrostatic field,
𝐸⃗⃗ =
𝑞 rˆ
4𝜋𝜖0 𝑟2
The divergence of this field is
∇⃗⃗. 𝐸⃗⃗ = ⃗∇⃗. (
𝑟
⃗
) = 𝑟3 𝑟3 𝑟3
3 3
− = 0
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3. NPTEL – Physics – Mathematical Physics - 1
Think for a moment. Is this a correct result? Let us clarify this point later.
Example
Consider a vector
𝐴⃗ = (𝑥 + 3𝑦) xˆ + (𝑦 − 2𝑧) yˆ + (𝑥 + 𝛼𝑧) zˆ
For what value of 𝛼, 𝐴⃗ is solenoidal (for which the divergence is zero).
Solution
𝑑𝑥 𝑑𝑦 𝑑𝑧
∇⃗⃗.𝐴⃗ = (𝑥̂ + yˆ + zˆ 𝑑
). [(𝑥 + 3𝑦) xˆ + (𝑦 − 2𝑧) yˆ + (𝑥 + 𝛼𝑧)
zˆ ]
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𝑑 𝑑
= 1+ 1 + 𝛼 = 0 ⟹ 𝛼 = −2