In my math courseguides (written by my professors), curl and divergence of vector fields are defined very precisely, but due to a lack of examples all is lost in transelation... I was able to figure out how to interpret divergence, but an interpretation of curl still eludes me. What does this vector represent? Can you give a decent example so I can remember what it is and apply it to other examples? All I know is that it has something to do with rotation. Best explanation and example gets the 10 points.
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Let's say that there is a moving stream of particles, in 2D or 3D, that can be described by a vector field, itself best described by a bunch of velocity direction arrows, i.e., the direction each particle is making at any given moment. If you take a certain closed area or region of space and added up the angular momenta of all the particles within the closed area or region, and if the net is not zero, then that's curl. You might not actually see a "whirlpool" going on in that area or region, but nonetheless, there's a potential for a "spinning solar system" to form inside it should the particles decide to coalesce. That's why this subject matter of vector curl can be confusing because of this, it doesn't necessarily involve some kind of a "center of whirling", only that the net angular momenta of the points in a closed area or region in a vector field is nonzero. And the curl vector is that axis of spin, even though you may not actually see exactly where it is spinning or what direction is the axis of spin. It could be that the only way you can tell is to "do the math".
Edit: A good example of the non-obviousness of curl is to imagine a smooth vector stream of parallel trajectories, like a straight river. But imagine that the flow varies across the width of the river. If you take one small area of the river that spans straight streams that are going at different rates, there will be a non-zero curl, even though there seems to be no "whirlpool" action going on.
Edit: A good example of the non-obviousness of curl is to imagine a smooth vector stream of parallel trajectories, like a straight river. But imagine that the flow varies across the width of the river. If you take one small area of the river that spans straight streams that are going at different rates, there will be a non-zero curl, even though there seems to be no "whirlpool" action going on.
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I feel your pain! Straighteners just damages your hair. Embrace your curls.