I have learned how to compute the two for a given vector field, but I don’t understand what the divergence and curl actually tells us.
In: 2
Divergence is how much vectors point away/towards a point, and curl is how much vectors circle around a point (measured as a vector pointing along the axis of that circle)
Imagine your vector field being a current of liquid flow. The divergence measures how much liquid is created in a point, I.E. if you have an area with a positive divergence more liquid is leaving it than entering it. Curl measures whirls in the current, I.E. if you move in a circle and get accelerated by the current then you are in an area with curl, if you move in a circle and overall your speed is unchanged (because it accelerated in one direction but stopped you on the way back) curl must be zero.
This is usefull in many physics applications. For example the electrical field is created by electrical charges. That means the divergence of the field is the charge density (new field lines always start at a charge, and end in a charge of the opposite sign)
Let’s start with a classic vector field: the velocity field of a fluid in 3D space.
Div at any point is positive if more of the local flow is away from the point than towards it. In other words div tells you if the fluid is expanding or being compressed at that point.
Curl tells you if there is overall rotation in the flow of the fluid at a point. Anywhere in a whirlpool you will see non-zero curl.
These visuals may help: https://books.physics.oregonstate.edu/GSF/visdivcurl.html
Divergence: how much more vectors point away from a point as they get farther from the point.
Curl: how much more vectors point in circles around a point as they get farther form the point.
Also see the bottom examples, where the “more” bit is missing and therefore thetres no divergence or curl.
Disclaimer: I’ve never learned calculus.
Edit: as for practical examples, an exploding firework has divergence, and a whirlpool in water has curl.
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