# Divergence

**An operation on a Vector Field which returns a scalar field representing the degree to which the field flows outwards.**

The **divergence** of a vector field  is a scalar field which describes the rate at which the vector field changes the volume of a region around a point as the region contracts to the point.

A physical interpretation of divergence is that it represents the amount that the vector field _flows outwards_ at every point. The terms "source" and "sink" are common ways to describe regions where the vector field originates or terminates and thus has positive or negative divergence, respectively. 

## Definition

### Cartesian coordinate definition

In general, the definition of divergence using [Cartesian coordinates](Cartesian%20coordinates.md) are more commonly used and simpler than the coordinate-less definition which is defined at a point.

Divergence

The _divergence_ of a vector field $\textbf{F} =F_1 e_1 +F_2 e_2 ... +F_N e_N$  or $F(x,y,z)=(Fx​,Fy​,Fz​)$; the scalar field  where is the sum its  partial derivatives: 

$$div\; F = \nabla \cdot F = \frac{\partial F_x}{\partial x}  +\frac{\partial F_z}{\partial z} + \frac{\partial F_y}{\partial y} $$ of the component function with respect to its axes. Note $\nabla F$ is **NORMAL to F**.

# Dot Product and Flux

## Flux through a Surface

$$
\text{Flux} = \mathbf{F} \cdot \mathbf{n}\, dS
$$

Where: - $\mathbf{F}$ = vector field
- $\mathbf{n}$ = surface normal
- $dS$ = surface element

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## Dot Product Expansion

$$
\mathbf{F} \cdot \mathbf{n} = |\mathbf{F}|\,|\mathbf{n}| \cos\theta
$$

Where: - $\theta$ = angle between the vector field and the surface
normal

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## Interpretation

Only the component of the vector field **parallel to the surface
normal** contributes to flow through the surface.

-   If $\theta = 0^\circ$:
    $$ \cos\theta = 1 $$
    Full flow through the surface
-   If $\theta = 90^\circ$:\
    $$ \cos\theta = 0 $$
    No flow through the surface
-   If $\theta > 90^\circ$:
    $$ \cos\theta < 0 $$
    Flow is **into** the surface

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## Key Insight

$$
\text{Dot product} = \text{projection of the vector field onto the normal}
$$

This projection represents the **actual flow crossing the surface**.

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## Link to Divergence

$$
\nabla \cdot \mathbf{F}
$$

Represents the **net flow out of a small volume**.

$$
\iiint (\nabla \cdot \mathbf{F})\, dV = \iint \mathbf{F} \cdot \mathbf{n}\, dS
$$

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## One-Line Summary

-   Dot product → flow **through** a surface
-   Divergence → net flow **out of** a volume
-   $\cos\theta$ → how much of the vector actually crosses the surface




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