6.5 Divergence and curl  (Page 3/9)

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

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

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 F on a region can be translated into a line integral of 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 v be a vector field modeling the velocity of a fluid. Since the divergence of v at point P measures the “outflowing-ness” of the fluid at P , $\text{div}\text{v}(P)>0$ implies that more fluid is flowing out of P than flowing in. Similarly, $\text{div}\text{v}(P)<0$ implies the more fluid is flowing in to P than is flowing out, and $\text{div}\text{v}(P)=0$ implies the same amount of fluid is flowing in as flowing out.

Determining flow of a fluid

Suppose $\text{v}(x,y)=⟨\text{−}xy,y⟩,y>0$ models the flow of a fluid. Is more fluid flowing into point $(1,4)$ than flowing out?

To determine whether more fluid is flowing into $(1,4)$ than is flowing out, we calculate the divergence of v at $(1,4)\text{:}$

$\text{div}\left(\text{v}\right)=\frac{\partial }{\partial x}\left(\text{−}xy\right)+\frac{\partial }{\partial y}\left(y\right)=\text{−}y+1.$

To find the divergence at $(1,4),$ substitute the point into the divergence: $\mathrm{-4}+1=\mathrm{-3}.$ Since the divergence of v at $(1,4)$ is negative, more fluid is flowing in than flowing out ( [link] ).

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Curl

The second operation on a vector field that we examine is the curl, which measures the extent of rotation of the field about a point. Suppose that F represents the velocity field of a fluid. Then, the curl of F at point P is a vector that measures the tendency of particles near P to rotate about the axis that points in the direction of this vector. The magnitude of the curl vector at P measures how quickly the particles rotate around this axis. In other words, the curl at a point is a measure of the vector field’s “spin” at that point. Visually, imagine placing a paddlewheel into a fluid at P , with the axis of the paddlewheel aligned with the curl vector ( [link] ). The curl measures the tendency of the paddlewheel to rotate.