Divergence is an operation on a vector field that tells us how the field behaves toward or away from a point. Locally, the divergence of a vector field ${\bf{F}}$ in $\mathbb{R}^2$ or $\mathbb{R}^3$ at a particular point $P$ is a measure of the “outflowing-ness” of the vector field at $P$. If ${\bf{F}}$ represents the velocity of a fluid, then the divergence of ${\bf{F}}$ at $P$ measures the net rate of change with respect to time of the amount of fluid flowing away from $P$ (the tendency of the fluid to flow “out of” $P$). In particular, if the amount of fluid flowing into P is the same as the amount flowing out, then the divergence at $P$ is zero.

By contrast, consider radial vector field ${\bf{R}}(x,y)=\langle-x,-y\rangle$ in Figure 2. At any given point, more fluid is flowing in than is flowing out, and therefore the “outgoingness” of the field is negative. We expect the divergence of this field to be negative, and this is indeed the case, as $\text{div }({\bf{R}})=\frac{\partial}{\partial{x}}(-x)+\frac{\partial}{\partial{y}}(-y)=-2$.

To get a global sense of what divergence is telling us, suppose that a vector field in $\mathbb{R}^2$ represents the velocity of a fluid. Imagine taking an elastic circle (a circle with a shape that can be changed by the vector field) and dropping it into a fluid. If the circle maintains its exact area as it flows through the fluid, then the divergence is zero. This would occur for both vector fields in Figure 1. On the other hand, if the circle’s shape is distorted so that its area shrinks or expands, then the divergence is not zero. Imagine dropping such an elastic circle into the radial vector field in Figure 2 so that the center of the circle lands at point $(3, 3)$. The circle would flow toward the origin, and as it did so the front of the circle would travel more slowly than the back, causing the circle to “scrunch” and lose area. This is how you can see a negative divergence.

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One application for divergence occurs in physics, when working with magnetic fields. A magnetic field is a vector field that models the influence of electric currents and magnetic materials. Physicists use divergence in Gauss’s law for magnetism, which states that if ${\bf{B}}$ is a magnetic field, then $\nabla\cdot{\bf{B}}=0$; in other words, the divergence of a magnetic field is zero.

Another application for divergence is detecting whether a field is source free. Recall that a source-free field is a vector field that has a stream function; equivalently, a source-free field is a field with a flux that is zero along any closed curve. The next two theorems say that, under certain conditions, source-free vector fields are precisely the vector fields with zero divergence.

Proof

Since ${\bf{F}}$ is source free, there is a function $g(x, y)$ with $g_y=P$ and $-g_x=Q$. Therefore, ${\bf{F}}=\langle{g}_y,-g_x\rangle$ and $\text{div }{\bf{F}}=g_{yx}-g_{xy}=0$ by Clairaut’s theorem.

$_\blacksquare$

The converse of Divergence of a Source-Free Vector Field Theorem is true on simply connected regions, but the proof is too technical to include here. Thus, we have the following theorem, which can test whether a vector field in $\mathbb{R}^2$ is source free.

Recall that the flux form of Green’s theorem says that

$\large{\displaystyle\oint_C{\bf{F}}\cdot{\bf{N}}ds=\displaystyle\iint_DP_x+Q_ydA}$,

where $C$ is a simple closed curve and $D$ is the region enclosed by $C$. Since $P_x+Q_y=\text{div }{\bf{F}}$, Green’s theorem is sometimes written as

$\large{\displaystyle\oint_C{\bf{F}}\cdot{\bf{N}}ds=\displaystyle\iint_D\text{div }{\bf{F}}dA}$.

Therefore, Green’s theorem can be written in terms of divergence. If we think of divergence as a derivative of sorts, then Green’s theorem says the “derivative” of ${\bf{F}}$ on a region can be translated into a line integral of ${\bf{F}}$ along the boundary of the region. This is analogous to the Fundamental Theorem of Calculus, in which the derivative of a function $f$ on a line segment $[a,b]$ can be translated into a statement about $f$ on the boundary of $[a,b]$. Using divergence, we can see that Green’s theorem is a higher-dimensional analog of the Fundamental Theorem of Calculus.

We can use all of what we have learned in the application of divergence. Let ${\bf{v}}$ be a vector field modeling the velocity of a fluid. Since the divergence of${\bf{v}}$ at point $P$ measures the “outflowing-ness” of the fluid at $P$, $\text{div }{\bf{v}}(P)>0$ implies that more fluid is flowing out of $P$ than flowing in. Similarly, $\text{div }{\bf{v}}(P)<0$ implies the more fluid is flowing in to $P$ than is flowing out, and $\text{div }{\bf{v}}(P)=0$ implies the same amount of fluid is flowing in as flowing out.