6.5 Divergence and curl  (Page 3/9)

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Determining whether a field is source free

Is field $\text{F}\left(x,y\right)=⟨{x}^{2}y,5-x{y}^{2}⟩$ source free?

Note the domain of F is ${ℝ}^{2},$ which is simply connected. Furthermore, F is continuous with differentiable component functions. Therefore, we can use [link] to analyze F . The divergence of F is

$\frac{\partial }{\partial x}\left({x}^{2}y\right)+\frac{\partial }{\partial y}\left(5-x{y}^{2}\right)=2xy-2xy=0.$

Therefore, F is source free by [link] .

Let $\text{F}\left(x,y\right)=⟨\text{−}ay,bx⟩$ be a rotational field where a and b are positive constants. Is F source free?

Yes

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

${\oint }_{C}\text{F}·\text{N}ds={\iint }_{D}{P}_{x}+{Q}_{y}dA,$

where C is a simple closed curve and D is the region enclosed by C . Since ${P}_{x}+{Q}_{y}=\text{div}\phantom{\rule{0.2em}{0ex}}\text{F},$ Green’s theorem is sometimes written as

${\oint }_{C}\text{F}·\text{N}ds={\iint }_{D}\text{div}\phantom{\rule{0.2em}{0ex}}\text{F}dA.$

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 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 $\left[a,b\right]$ can be translated into a statement about $f$ on the boundary of $\left[a,b\right].$ 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 , $\text{div}\phantom{\rule{0.2em}{0ex}}\text{v}\left(P\right)>0$ implies that more fluid is flowing out of P than flowing in. Similarly, $\text{div}\phantom{\rule{0.2em}{0ex}}\text{v}\left(P\right)<0$ implies the more fluid is flowing in to P than is flowing out, and $\text{div}\phantom{\rule{0.2em}{0ex}}\text{v}\left(P\right)=0$ implies the same amount of fluid is flowing in as flowing out.

Determining flow of a fluid

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

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

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

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

For vector field $\text{v}\left(x,y\right)=⟨\text{−}xy,y⟩,y>0,$ find all points P such that the amount of fluid flowing in to P equals the amount of fluid flowing out of P .

All points on line $y=1.$

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.

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20/(×-6^2)
Salomon
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Salomon
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Salomon
I got X =-6
Salomon
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