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The '''divergence''' is a way of expressing a certain type of [[derivative]] of a [[vector field]]. It is typically a field of 3-dimensional vectors defined 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.
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".
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".
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''.
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Physics]]
[[Category:Physics]]