notes/DIV.md

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 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} 

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

---

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

---

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

---

Key Insight

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

This projection represents the actual flow crossing the surface.

---

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

---

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

---