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divergence (Definition)

Divergence

The divergence of a vector field is defined as

$\displaystyle \nabla \cdot {\bf V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z}$

This is easily seen from the definition of the dot product and that of the del operator

$\displaystyle \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z $

$\displaystyle \nabla = \frac{\partial}{\partial x} {\bf\hat{i}} + \frac{\partial}{\partial y}{\bf\hat{j}} + \frac{\partial}{\partial z}{\bf\hat{z}}$

carrying out the dot product with ${\bf V}$ then gives (1).

Physical Meaning

(this section is a work in progress)

Building physical intuition about the divergence of a vector field can be gained by considering the flow of a fluid. One of the most simple vector fields is a uniform velocity field shown in below figure.

Figure: Uniform Flow
\includegraphics[scale=1]{UniformFlow.eps}

Mathematically, this field would be

$\displaystyle {\bf V} = 5 {\bf\hat{i}} $

The divergence is then

$\displaystyle \nabla \cdot {\bf V} = \frac{\partial}{\partial x} 5 = 0 $

Source/Sink flow field ( div > 0 / div < 0)

Figure: Positive Divergence
\includegraphics[scale=1]{PositiveDivergence.eps}
Figure: Negative Divergence
\includegraphics[scale=1]{NegativeDivergence.eps}

Circular flow with zero divergence

Figure: Circular Flow
\includegraphics[scale=1]{CircularFlow.eps}

Coordinate Systems

Cartesian Coordinates

$\displaystyle \nabla \cdot {\bf V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z}$

Cylindrical Coordinates

$\displaystyle \nabla \cdot {\bf V} = \frac{1}{r}\frac{\partial}{\partial r} (r ... ...} \frac{\partial V_{\theta}}{\partial \theta} + \frac{\partial V_z}{\partial z}$

Spherical Coordinates

$\displaystyle \nabla \cdot {\bf V} = \frac{1}{r^2}\frac{\partial}{\partial r} (... ...ta} sin \theta) + \frac{1}{r sin \theta}\frac{\partial V_{\phi}}{\partial \phi}$



"divergence" is owned by bloftin.

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See Also: curl, gradient, Gauss's Law, gradient

Other names:  divergence of a vector field

Attachments:
sources and sinks of vector field (Topic) by pahio

Cross-references: field, velocity, vector fields, work, section, operator, dot product
There are 7 references to this object.

This is version 2 of divergence, born on 2006-08-28, modified 2006-08-29.
Object id is 221, canonical name is Divergence.
Accessed 2363 times total.

Classification:
Physics Classification: 02. (Mathematical methods in physics)
Pending Errata and Addenda
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