# Finding Scalar Curl and Divergence from a Picture of Vector Field

## Homework Statement:

Which of the following is the divergence of the vector field shown (see attachment for visual)?
f(x,y)=
a) x
b) y
c) -x
d) -y
e) x+y
f) x-y
g) y-x
h) -y-x

## Relevant Equations:

div(f) = d/dx f1 + d/dy f2 + d/dz f3
scalar curl of f = d f2/dx - d f1/dy
For divergence: We learned to draw a circle at different locations and to see if gas is expanding/contracting. Whenever the y-coordinate is positive, the gas seems to be expanding, and it's contracting when negative. I find it hard to tell if the gas is expanding or contracting as I go to the right and left, so I'm not sure what x-dependence it has. It seems like it's symmetrical, so perhaps no x-dependence and f(x,y)=y?

For curl, I'm kind of confused. I thought scalar curl had to do with rotation. But some examples the teacher gives just has straight lines (doesn't seem to be rotation) and there is a non-zero scalar curl. It does seem to be rotating clockwise on the right side and counterclockwise on the left, so does that just mean it's equal to -x?

I'm struggling for better intuition here and not sure if I'm just reaching.

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Also, for curl, we sometimes were told to plug in points, but I seem to be getting an inconsistency. I tried:
20,10: negative (because tends toward clockwise)
10, 20: positive
-10,20: negative
-20,10: positive
-20,-10: positive
-10,-20: negative
10, -20: positive
20, -10: negative

The only one that seemed promising was y-x. I'm thinking it must have x and y dependence since one quadrant can be both positive and negative, depending on the point. But y-x fails (10, -20), because that would be negative but the graph has CCW rotation.