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Determining whether a field is source free

Is field F ( x , y ) = x 2 y , 5 x y 2 source free?

Note the domain of F is 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

x ( x 2 y ) + y ( 5 x y 2 ) = 2 x y 2 x y = 0 .

Therefore, F is source free by [link] .

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Let F ( x , y ) = a y , b x be a rotational field where a and b are positive constants. Is F source free?


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

C F · N d s = D P x + Q y d A ,

where C is a simple closed curve and D is the region enclosed by C . Since P x + Q y = div F , Green’s theorem is sometimes written as

C F · N d s = D div F d A .

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 , div v ( P ) > 0 implies that more fluid is flowing out of P than flowing in. Similarly, div v ( P ) < 0 implies the more fluid is flowing in to P than is flowing out, and div v ( P ) = 0 implies the same amount of fluid is flowing in as flowing out.

Determining flow of a fluid

Suppose v ( x , y ) = x y , 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 ) :

div ( v ) = x ( x y ) + y ( y ) = y + 1 .

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

A vector field in two dimensions with negative divergence at (1,4). The arrows are very flat but become more vertical closer to the y axis. Above the x axis, the arrows point up and towards the y axis on either side of it. Below the x axis, the arrows point down and away from the y axis on either side of it.
Vector field v ( x , y ) = x y , y has negative divergence at ( 1 , 4 ) .
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For vector field v ( x , y ) = x y , 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 .

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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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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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