# 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.

#### Questions & Answers

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sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
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I got X =-6
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ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
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a perfect square v²+2v+_
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or infinite solutions?
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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Differences Between Laspeyres and Paasche Indices
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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