6.5 Divergence and curl  (Page 2/9)

To get a global sense of what divergence is telling us, suppose that a vector field in ${ℝ}^2$ 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 [link] . 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 [link] 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.

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 B is a magnetic field, then $\nabla ·\text{B}=0;$ in other words, the divergence of a magnetic field is zero.

Determining whether a field is magnetic

Is it possible for $\text{F}\left(x,y\right)=⟨x^{2}y,y-x{y}^2⟩$ to be a magnetic field?

If F were magnetic, then its divergence would be zero. The divergence of F is

$\frac{\partial }{\partial x}\left(x^{2}y\right)+\frac{\partial }{\partial y}\left(y-x{y}^2\right)=2xy+1-2xy=1$

and therefore F cannot model a magnetic field ( [link] ).

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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 F is source free, there is a function $g\left(x,y\right)$ with $g_y=P$ and $\text{−}{g}_x=Q.$ Therefore, $\text{F}=⟨g_y,\text{−}{g}_x⟩$ and $\text{div}\text{F}=g_{yx}-g_{xy}=0$ by Clairaut’s theorem.

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The converse of [link] 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 ${ℝ}^2$ is source free.