Learning Objectives
- Determine divergence from the formula for a given vector field.
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 [latex]{\bf{F}}[/latex] in [latex]\mathbb{R}^2[/latex] or [latex]\mathbb{R}^3[/latex] at a particular point [latex]P[/latex] is a measure of the “outflowing-ness” of the vector field at [latex]P[/latex]. If [latex]{\bf{F}}[/latex] represents the velocity of a fluid, then the divergence of [latex]{\bf{F}}[/latex] at [latex]P[/latex] measures the net rate of change with respect to time of the amount of fluid flowing away from [latex]P[/latex] (the tendency of the fluid to flow “out of” [latex]P[/latex]). In particular, if the amount of fluid flowing into P is the same as the amount flowing out, then the divergence at [latex]P[/latex] is zero.
definition
If [latex]{\bf{F}}\langle{P},Q,R\rangle[/latex] is a vector field in [latex]\mathbb{R}^3[/latex] and [latex]P_x[/latex], [latex]Q_y[/latex], and [latex]R_z[/latex] all exist, then the divergence of [latex]{\bf{F}}[/latex] is defined by
[latex]\large{\text{div }{\bf{F}}=P_x+Q_y+R_z=\frac{\partial{P}}{\partial{x}}+\frac{\partial{Q}}{\partial{y}}+\frac{\partial{R}}{\partial{z}}}[/latex].
Note the divergence of a vector field is not a vector field, but a scalar function. In terms of the gradient operator [latex]\nabla=\left\langle\frac{\partial}{\partial{x}},\frac{\partial}{\partial{y}},\frac{\partial}{\partial{z}}\right\rangle[/latex], divergence can be written symbolically as the dot product
[latex]\large{\text{div }{\bf{F}}=\nabla\cdot{\bf{F}}}[/latex].
Note this is merely helpful notation, because the dot product of a vector of operators and a vector of functions is not meaningfully defined given our current definition of dot product.
If [latex]{\bf{F}}\langle{P},Q\rangle[/latex] is a vector field in [latex]\mathbb{R}^2[/latex], and [latex]P_x[/latex] and [latex]Q_y[/latex] both exist, then the divergence of [latex]{\bf{F}}[/latex] is defined similarly as
[latex]\large{\text{div }{\bf{F}}=P_x+Q_y=\frac{\partial{P}}{\partial{x}}+\frac{\partial{Q}}{\partial{y}}=\nabla\cdot{\bf{F}}}[/latex].
To illustrate this point, consider the two vector fields in Figure 1. At any particular point, the amount flowing in is the same as the amount flowing out, so at every point the “outflowing-ness” of the field is zero. Therefore, we expect the divergence of both fields to be zero, and this is indeed the case, as
[latex]\large{\text{div }(\langle1,2\rangle)=\frac{\partial}{\partial{x}}(1)+\frac{\partial}{\partial{y}}(2)=0\text{ and div}(\langle-y,x\rangle)=\frac{\partial}{\partial{x}}(-y)+\frac{\partial}{\partial{y}}(x)=0}[/latex].
Example: calculating divergence at a Point
If [latex]{\bf{F}}(x,y,z)=e^x{\bf{i}}+yz{\bf{j}}-y^2{\bf{k}}[/latex], then find the divergence of [latex]{\bf{F}}[/latex] at [latex](0, 2, -1)[/latex].
try it
Find [latex]\text{div }{\bf{F}}[/latex] for [latex]{\bf{F}}(x,y,z)=\langle{x}y,5-z^2y,x^2+y^2\rangle[/latex].
Watch the following video to see the worked solution to the above Try It
Example: determining whether a field is magnetic
Is it possible for [latex]{\bf{F}}(x,y)=\langle{x}^2y,y-xy^2\rangle[/latex] to be a magnetic field?
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.
theorem: divergence of a source-free vector field
If [latex]{\bf{F}}=\langle{P},Q\rangle[/latex] is a source-free continuous vector field with differentiable component functions, then [latex]\text{div }{\bf{F}}=0[/latex].
Proof
Since [latex]{\bf{F}}[/latex] is source free, there is a function [latex]g(x, y)[/latex] with [latex]g_y=P[/latex] and [latex]-g_x=Q[/latex]. Therefore, [latex]{\bf{F}}=\langle{g}_y,-g_x\rangle[/latex] and [latex]\text{div }{\bf{F}}=g_{yx}-g_{xy}=0[/latex] by Clairaut’s theorem.
[latex]_\blacksquare[/latex]
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 [latex]\mathbb{R}^2[/latex] is source free.
Theorem: divergence test for source-free vector fields
Let [latex]{\bf{F}}=\langle{P},Q\rangle[/latex] be a continuous vector field with differentiable component functions with a domain that is simply connected. Then, [latex]\text{div }{\bf{F}}=0[/latex] if and only if [latex]{\bf{F}}[/latex] is source free.
Example: determining whether a field is Source free
Is field [latex]{\bf{F}}(x,t)=\langle{x}^2y,5-xy^2\rangle[/latex] source free?
try it
Let [latex]{\bf{F}}(x,y)=\langle-ay,bx\rangle[/latex] be a rotational field where [latex]a[/latex] and [latex]b[/latex] are positive constants. Is [latex]{\bf{F}}[/latex] source free?
Recall that the flux form of Green’s theorem says that
[latex]\large{\displaystyle\oint_C{\bf{F}}\cdot{\bf{N}}ds=\displaystyle\iint_DP_x+Q_ydA}[/latex],
where [latex]C[/latex] is a simple closed curve and [latex]D[/latex] is the region enclosed by [latex]C[/latex]. Since [latex]P_x+Q_y=\text{div }{\bf{F}}[/latex], Green’s theorem is sometimes written as
[latex]\large{\displaystyle\oint_C{\bf{F}}\cdot{\bf{N}}ds=\displaystyle\iint_D\text{div }{\bf{F}}dA}[/latex].
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 [latex]{\bf{F}}[/latex] on a region can be translated into a line integral of [latex]{\bf{F}}[/latex] along the boundary of the region. This is analogous to the Fundamental Theorem of Calculus, in which the derivative of a function [latex]f[/latex] on a line segment [latex][a,b][/latex] can be translated into a statement about [latex]f[/latex] on the boundary of [latex][a,b][/latex]. 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 [latex]{\bf{v}}[/latex] be a vector field modeling the velocity of a fluid. Since the divergence of[latex]{\bf{v}}[/latex] at point [latex]P[/latex] measures the “outflowing-ness” of the fluid at [latex]P[/latex], [latex]\text{div }{\bf{v}}(P)>0[/latex] implies that more fluid is flowing out of [latex]P[/latex] than flowing in. Similarly, [latex]\text{div }{\bf{v}}(P)<0[/latex] implies the more fluid is flowing in to [latex]P[/latex] than is flowing out, and [latex]\text{div }{\bf{v}}(P)=0[/latex] implies the same amount of fluid is flowing in as flowing out.
Example: determining the flow of a fluid
Suppose [latex]{\bf{v}}(x,y)=\langle-xy,y\rangle[/latex], [latex]y>0[/latex] models the flow of a fluid. Is more fluid flowing into point [latex](1, 4)[/latex] than flowing out?
try it
For vector field [latex]{\bf{v}}(x,y)=\langle-xy,y\rangle[/latex], [latex]y>0[/latex] find all points [latex]P[/latex] such that the amount of fluid flowing in to [latex]P[/latex] equals the amount of fluid flowing out of [latex]P[/latex].