98 lines
2.9 KiB
Markdown
98 lines
2.9 KiB
Markdown
# Divergence
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**An operation on a Vector Field which returns a scalar field representing the degree to which the field flows outwards.**
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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.
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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.
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## Definition
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### Cartesian coordinate definition
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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.
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Divergence
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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:
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$$div\; F = \nabla \cdot F = \frac{\partial F_x}{\partial x} +\frac{\partial F_z}{\partial z} + \frac{\partial F_y}{\partial y} $$ of the component function with respect to its axes. Note $\nabla F$ is **NORMAL to F**.
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# Dot Product and Flux
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## Flux through a Surface
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$$
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\text{Flux} = \mathbf{F} \cdot \mathbf{n}\, dS
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$$
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Where: - $\mathbf{F}$ = vector field
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- $\mathbf{n}$ = surface normal
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- $dS$ = surface element
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------------------------------------------------------------------------
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## Dot Product Expansion
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$$
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\mathbf{F} \cdot \mathbf{n} = |\mathbf{F}|\,|\mathbf{n}| \cos\theta
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$$
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Where: - $\theta$ = angle between the vector field and the surface
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normal
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------------------------------------------------------------------------
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## Interpretation
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Only the component of the vector field **parallel to the surface
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normal** contributes to flow through the surface.
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- If $\theta = 0^\circ$:
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$$ \cos\theta = 1 $$
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Full flow through the surface
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- If $\theta = 90^\circ$:\
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$$ \cos\theta = 0 $$
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No flow through the surface
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- If $\theta > 90^\circ$:
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$$ \cos\theta < 0 $$
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Flow is **into** the surface
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------------------------------------------------------------------------
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## Key Insight
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$$
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\text{Dot product} = \text{projection of the vector field onto the normal}
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$$
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This projection represents the **actual flow crossing the surface**.
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------------------------------------------------------------------------
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## Link to Divergence
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$$
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\nabla \cdot \mathbf{F}
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$$
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Represents the **net flow out of a small volume**.
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$$
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\iiint (\nabla \cdot \mathbf{F})\, dV = \iint \mathbf{F} \cdot \mathbf{n}\, dS
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$$
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------------------------------------------------------------------------
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## One-Line Summary
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- Dot product → flow **through** a surface
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- Divergence → net flow **out of** a volume
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- $\cos\theta$ → how much of the vector actually crosses the surface
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