# Divergence of a Vector Field and its Sources and Sinks

Level 3 (with higher mathematics)
Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.
Updated by Alexander Fufaev on

Divergence $$\nabla \cdot \boldsymbol{F}$$ of a vector field $$\boldsymbol{F}$$ is defined as the scalar product between the nabla operator $$\nabla$$ and the vector field $$\boldsymbol{F}$$:

Definition: Divergence of a vector field
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Here $$F_{\text x}$$ is the first, $$F_{\text y}$$ second and $$F_{\text z}$$ the third component of the following three-dimensional vector field $$\boldsymbol{F}(x,y,z)$$:

Three dimensional vector field
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As discussed in the lesson on Maxwell's equations, the vector field $$\boldsymbol{F}$$ can represent, for example, the electric field $$\boldsymbol{E}$$ or the magnetic field $$\boldsymbol{B}$$.

The result $$\nabla \cdot \boldsymbol{F}$$ of the divergence in Eq. 1 is a scalar function (no vector quantity anymore)! So, if a concrete place is inserted for $$(x,y,z)$$, then the scalar function results in a usual number: $$\nabla \cdot \boldsymbol{F}(x,y,z)$$. This number is a measure for the divergence of the vector field at the considered location $$(x,y,z)$$. It can be a positive or negative number or even zero. Depending on whether the number is positive, negative or zero, it has a different physical meaning.

## Positive divergence - source of a vector field

We assume that we have inserted a concrete location $$(x,y,z)$$ into the divergence field $$\nabla \cdot \boldsymbol{F}(x,y,z)$$ and obtained a positive number:

Divergence is positive
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Then the considered point $$(x,y,z)$$ in space is a source of the vector field $$\boldsymbol{F}$$. If this location is enclosed with an arbitrary surface (e.g. with a cube surface), then the flux of the vector field through this surface is also positive. The vector field points out of the surface.

## Negative divergence - sink of a vector field

Now assume that we have obtained a negative number after inserting a concrete location into the divergence field:

Divergence is negative
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Then the considered location $$(x,y,z)$$ is a sink of the vector field $$\boldsymbol{F}$$. If this place is enclosed with an arbitrary surface, then the flux of the vector field through this surface is also negative. The vector field goes into the surface.

## Divergence is zero - divergence-free vector field

Now assume that we have obtained zero after inserting a concrete location into the divergence field:

Divergence is zero
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Then at the location $$(x,y,z)$$ the considered vector field is divergence-free. That means: If this point is enclosed with any surface, then the flux of the vector field through this surface is also zero. The vector field does not point into this surface, but also does not point out. Or there is just as much vector field pointing into the surface as out, so that the two opposite contributions cancel each other out and the divergence is net zero.

In the examples of negative divergence and divergence-free fields, we got out a constant divergence that is independent of the location. But the divergence field $$\nabla \cdot \boldsymbol{F}$$ must not necessarily result in a constant! They were just simple examples. In general the divergence field depends on all three coordinates: $$\nabla \cdot \boldsymbol{F}(x,y,z)$$.

Now you should know how divergence of a vector field is calculated and physically interpreted. This knowledge will help you grasp the Gauss integral theorem, which is incredibly important for understanding Maxwell's equations.