Matching equations to correct plot

Question

Should match each divergence and curl to correct plot but cannot insert the images here so a description of what the plot should look like will be helpful then I can match to the correct plot.

1.) What will the plots look like for these 'divergence':

1. 18x - 56y - 41

1. 0

1. 6

2.) What will the plots look like for these 'curl', where constant value z=0 for these plots. (should be vector columns but don't know how to write vector columns here).

1. (0 0 0)
2. (0 0 8)
3. (0 0 7)

Answer 1

If you think of a vector field as a flow of some fluid then the divergence describes the rate at which the fluid is entering/leaving a small region about a fixed point. This is the idea you'll need to be able to make sense of this problem.

So how do you connect this very qualitative idea about divergence with specific functions? Well one way would be to try to imagine some vector fields that would have the divergence listed and then plot it (I recommend this approach to reinforce the idea.) For example: the vector field F(x,y) = <-x, y> will have divergence div(F) = -1 + 1 = 0. So if you draw a collection of arrows for F you can see what a field with divergence zero looks like. What you should see is that if you pick a point and then draw a closed shape around it (like a square or a circle) the vectors going 'into the shape' and 'out of the shape' from the boundary roughly add up to zero. In reality there are infinitely many vectors to check and we would use a line integral to see how it all adds up, but you may not have learned about that yet.

The end result is a vector field where the magnitudes will change size as you move away from the origin but in such a way that the 'vectors in' versus 'vectors out' of a point stay in balance--that is: the magnitudes grow in a "controlled" way.

When the divergence is 6 we have a net positive flow of the fluid out of a point (or a small shape around a given point) so the vectors leaving a small region represent about 6 times as much fluid flow as the fluid entering the same small region. You can see this again by making a very simple choice: F(x,y) = <6x, 0> which is simple to plot. Try drawing a square with side lengths 1 and line up the bottom edge with the x-axis; you'll see the vectors will line up with the top and bottom edge of the square so that nothing is flowing 'in' or 'out' of the tops, but only into the square from the left and out of the square from the right. Notice that the vectors corresponding to points on the right edge are six times the size of those on the left edge (assuming you put this square on the positive x-axis.) This is what you should look for in the divergence = 6 case: the vectors grow a bit faster as you move around the plane but that growth is still uniform.

This leaves the last divergence which you should now be able to reason about using the above examples. This 'flux' into and out of points now changes as you move around the x-y plane.

As for the curl we are looking for fluid rotation around point. We now get a vector since rotation requires an axis about which it is measured. The first field would have 'no rotation' while the other two would rotate counter-clockwise at two different rates (the bigger the magnitude of the curl the faster the rotation.)

Hopefully all of this is enough to get an idea of which graphs to choose. Follow-up with a comment if you have further questions!