Divergence

The divergence of a vector field is a scalar field that indicates at each point how much the vectors diverge (Latin divergere) in a small neighborhood of the point. If one interprets the vector field as the flow field of a quantity for which the continuity equation holds, then the divergence is the source density. Sinks have negative divergence. If the divergence is equal to zero everywhere, the field is called source-free.

The divergence is obtained from the vector field by applying a differential operator. Related differential operators yield the rotation of a vector field and the gradient of a scalar field. The mathematical field is vector analysis.

In physics, divergence is used, for example, in the formulation of Maxwell's equations or the various continuity equations. In the Ricci calculus, the quantity formed with the help of the covariant derivative is {\displaystyle \nabla _{k}T^{ik}}sometimes somewhat inaccurately called the divergence of a tensor {\displaystyle T^{ik}}(for this quantity, for example, the Gaussian integral theorem does not apply on curved manifolds).

Example from physics

For example, consider a still water surface that is struck by a thin stream of oil. The motion of the oil on the surface can be described by a two-dimensional (time-dependent) vector field: At any point in time, the flow velocity of the oil is given in the form of a vector. The point where the jet meets the water surface is a "source of oil", since oil flows away from there without any inflow on the surface. The divergence at this point is positive. In contrast, a place where the oil flows away from the water basin at the edge, for example, is called a sink. The divergence at this point is negative.

Definition

Let {\displaystyle {\vec {F}}\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n},(x_{1},\dots ,x_{n})\mapsto (F^{1},\dots ,F^{n})}a differentiable vector field. Then the divergence of {\vec {F}}is defined as.

{\displaystyle \operatorname {div} {\vec {F}}:=\nabla \cdot {\vec {F}}=({\tfrac {\partial }{\partial x_{1}}},\dotsc ,{\tfrac {\partial }{\partial x_{n}}})\cdot \left(F^{1},\dotsc ,F^{n}\right)={\tfrac {\partial }{\partial x_{1}}}F^{1}+\dotsb +{\tfrac {\partial }{\partial x_{n}}}F^{n}}

The divergence is the scalar product of the Nabla operator \nabla with the vector field {\vec {F}}.

Divergence is an operator on a vector field that results in a scalar field:

{\displaystyle \operatorname {div} [\cdot ]\ \colon C^{1}\left(\mathbb {R} ^{n},\mathbb {R} ^{n}\right)\to C^{0}\left(\mathbb {R} ^{n},\mathbb {R} \right)}

For the case of a three-dimensional vector field , the divergence in Cartesian coordinates is defined as\vec F(x_1,x_2,x_3).

{\displaystyle {\begin{matrix}\operatorname {div} \colon &C^{1}\left(\mathbb {R} ^{3},\mathbb {R} ^{3}\right)&\to &C^{0}\left(\mathbb {R} ^{3},\mathbb {R} \right)\\&{\vec {F}}=\left(F^{1},F^{2},F^{3}\right)&\mapsto &{\frac {\partial }{\partial x_{1}}}F^{1}+{\frac {\partial }{\partial x_{2}}}F^{2}+{\frac {\partial }{\partial x_{3}}}F^{3}\end{matrix}}}.

When writing {\displaystyle \nabla \cdot {\vec {F}}} it is important to write the multiplication point between \nabla and the vector field , {\vec {F}}since the \nabla operator would otherwise be understood as a gradient of the vector components (written {\displaystyle \nabla {\vec {F}}}).


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