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<small>For divergence of a series, see [[Convergence]]</small>

<small>For divergence of a series, see [[Convergence]]</small>

The '''divergence''' is a way of expressing a certain type of [[derivative]] of a [[vector field]].  It is typically defined for fields of 3-dimensional vectors on 3-dimensional space, but other dimensions are possible.  The divergence of a vector field is a [[scalar field]], that is, just a number at each point in space.

The '''divergence''' is a way of expressing a certain type of [[derivative]] of a [[vector field]].  It is typically defined for fields of 3-dimensional vectors on 3-dimensional space, but other dimensions are possible.  The divergence of a vector field is a [[scalar field]], that is, just a number at each point in space.  Vector fields with a divergence of zero are called ''divergenless'' or ''solenoidal''.

The divergence is written as though it were the [[dot product]] of the special symbol "<math>\nabla</math>" (which is commonly called "del" or "nabla"), with the given vector field, like this: <math>\nabla \cdot \vec V</math>.  This is usually pronounced "div V" or "del dot V".

Strictly speaking, The divergence of a Vector Field '''F''' is defined as the limit of the surface integral as the volume shrinks to 0:

In ordinary [[Cartesian coordinates]], the divergence is calculated as:

:<math>\operatorname{div}\,\mathbf{F} = \lim_{V \rightarrow 0} \frac{ \oint_\mathbf{S} \mathbf{F} \cdot d\mathbf{a} }{V}</math>

:<math>\nabla \cdot \vec V = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z}</math>

or, using suitable notation,

 

:<math>\nabla \cdot \vec V = \sum_{i=1}^3 \frac{\partial V_i}{\partial x_i}</math>

 

 

:<math>\operatorname{div}\,\mathbf{F} = \nabla \cdot \vec V = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z}</math>

or, using summation notation to extend it for n-dimension,

:<math>\nabla \cdot \vec V = \sum_{i=1}^n \frac{\partial V_i}{\partial x_i}</math>

If one thinks of <math>\nabla</math> as being a fictional vector field with components <math>(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})</math>, one can sort of see that the dot product notation makes sense.  This is also useful for remembering how to calculate a divergence.

If one thinks of <math>\nabla</math> as being a fictional vector field with components <math>(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})</math>, one can sort of see that the dot product notation makes sense.  This is also useful for remembering how to calculate a divergence.

Intuitively, the divergence measures the degree to which the vector field is diverging from a given point.  If you were to measure the divergence of the vector field of wind speed in the vicinity of a meteorological high pressure area, it would be positive, because the net motion of air is outward.  If measured near a low pressure area, the divergence would be negative.

Intuitively, the divergence measures the degree to which the vector field is diverging from a given point.  If you were to measure the divergence of the vector field of wind speed in the vicinity of a meteorological high pressure area, it would be positive, because the net motion of air is outward.  If measured near a low pressure area, the divergence would be negative.

 

Vector fields with a divergence of zero are called ''solenoidal''.

*[http://mathworld.wolfram.com/Divergence.html Divergence] from Wolfram Mathworld

[[Category:Calculus]]

[[Category:Calculus]]

[[Category:Physics]]

[[Category:Physics]]