Explain whether the divergence and curl of each of the vector fields
shown below are zero throught the entire region shown. Justify your answer.https://sphotos-a-ord.xx.fbcdn.net/hphotos-prn2/1185774_4956047513788_517908639_n.jpg [Broken]
The Attempt at a Solution
Ok, here's my attempt at answering for each field pictured.
A: Curl: The curl is not zero everywhere. Using the 'tiny paddlewheel test', there are points in the field (ie in the right half of the region shown) where the torque from the fluid on the left side (clockwise) is greater than the torque from the other side (counterclockwise), which would spin the paddlewheel, giving a non-zero curl. Divergence: It is zero everywhere. The reason is that there is no point at which there is a net inflow or net outflow of the field vectors, no point they 'spread away from' or 'converge upon.'
B: Curl: It is not zero everywhere in this region. Since the field vectors are longer (greater magnitude) nearer the 'center of the vortex' (not pictured), the torques on the near and far sides (relative to the center) are unequal, resulting in rotation of the tiny paddlewheel, i.e. nonzero curl. Divergence: The divergence is zero everywhere, since there are no 'sources' or 'sinks' of the field.
C: Curl: Zero everywhere, since there is no point that would cause a paddlewheel to rotate. Divergence: Not zero everywhere. In fact, the divergence is positive everywhere, since at any point the net outflow of the field exceeds the net inflow (the vectors get larger as we go farther along in the direction of the flow).
D: Curl: Zero everywhere. At every point, the clockwise torque equals the counterclockwise torque on the imaginary tiny paddlewheel, so it wouldn't rotate, therefore no curl (note this would not be the case at the unpictured center, where the wheel would rotate). Divergence: Zero everywhere, since for every point, each component of inward flow is canceled by an equal component of outward flow.
E: Curl: Not zero everywhere. A paddlewheel inserted into the center would spin clockwise. Divergence: Not zero everywhere, since at a point in the bottom left-hand corner, there is net inflow, giving negative divergence.
F: Curl: The curl is zero everywhere, since every clockwise torque on a paddlewheel would be canceled by a counterclockwise torque. This is easy to see in the center, but it appears to hold for every point in the region. Divergence: Not zero everywhere, since it appears to be negative at the center, since net inflow to a point located there exceeds net outflow.
Alright, that's my best shot. I'm not entirely confident in some of these, particularly the divergences. Can anyone spot any mistakes in my answers and, more importantly, in my way of thinking about this? I don't want to rely too much on the paddlewheel or 'source/sink' analogy, but since these are just pictures without numbers or anything, I feel like it gets me pretty far.
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