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I have been reviewing my vector calculus coursework as of late, and this time around, I've been really trying to understand the concepts intuitively/visually instead of just the math. Unfortunately, the identity div(curl F)=0 is giving me trouble.

I understand divergence is a measure of a vector field's compressibility. I understand curl is a vector field, representing F's rate of rotation. What I'm having a hard time visualizing is why curl F always produces a vector field that is incompressible?

Thanks for any help you can provide!