How are the formal definitions of curl interpreted? Where do they appear from and how to derive them?

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So here are formal definitions of curl for 2dimensional and 3dimensional. Can someone please explain how to derive these and interpret them? I am familiar with limit definition of curl, however these seem difficult to me. I know how to derive 3dimensional curl from 2 dimensions so I would rather need 2 dimensional curl explained as if i understand where 2dimensional formula appears from i can understand the 3rd dimension myself. Please.