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.
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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)
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