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ShadyPastry

What's the geometric interpretation of the sign/magnitude of divergence and curl?

motoole2

This is a good question. Last week, we studied the geometric meaning of the magnitude & direction (including the sign) of a vector. This past class covered the geometric meaning of divergence & curl. So it's just a matter of combining these two visual interpretations! So let me throw this question right back at you; based on what we have learnt in past lectures, any guesses what it means to take the sign/magnitude of the divergence/curl of a function?

uhhh

Divergence in 3D intuitively can still be the amount of some fluid going into/out of a point. I'm stuck on the 3D interpretation for curl though.

ShadyPastry

I would guess that the sign corresponds to in/out for divergence and clockwise/counterclockwise for curl, whereas the magnitude corresponds to, well, the magnitude of the flow/rotation.

Is that close?

motoole2

As uhhh mentioned, divergence operator measures the degree to which a vector field goes in/out of a point. The curl operator describes rotation of the vector field (e.g., in the slide above, the amount of rotation is highest/lowest at the center of each vortex).

And ShadyPastry got it spot on! The sign corresponds to in/out for divergence, or clockwise/counterclockwise for the curl operator.

Note that, in 2D, both divergence and curl operators produce a scalar value. In 3D, the divergence still results in a scalar value. However, the curl will produce a 3D vector, where each component of this vector represents the rotational component that is parallel to one of the yz-, xz-, or xy-planes.

If divergence and curl are still confusing, I found this article helpful.