Divergence

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For divergence of a series, see Convergence
\frac{d}{dx} \sin x=?\, This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

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 incompressible or solenoidal.

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

\operatorname{div}\,\mathbf{F} = \lim_{V \rightarrow 0} \frac{ \oint_\mathbf{S} \mathbf{F} \cdot d\mathbf{a} }{V}

Where the integral is over the boundary surface \mathbf{S}=\partial V surrounding the volume element V, which is taken to be zero in the limit.

Cartesian coordinates

For Cartesian coordinates, the divergence is written as though it were the dot product of the special symbol "\nabla" (which is commonly called "del" or "nabla"), with the given vector field, like this: \nabla \cdot \vec V. This is usually pronounced "div V" or "del dot V".

In ordinary 3-dimensional Cartesian coordinates, the divergence is calculated as:

\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}

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

\nabla \cdot \vec V = \sum_{i=1}^n \frac{\partial V_i}{\partial x_i}

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

The divergence is a true vector field operation—the result is independent of the coordinate system that is used. The proof of that, and its ramifications, are beyond the scope of this page.

The divergence is an extremely important operation in physics, mathematics, and engineering. It is perhaps most famous for its appearance in Maxwell's Equations.

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.

References

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