Is there a relationship between the Laplacian of a bivariate function and an iterated integral describing its "under-the-curve" volume within a certain domain and range?

Just curious...

If there is no such relationship, as is probably the case, could someone tell me what the purpose of the Laplacian, in multivariate functions,

Or, more generally, what the purpose of divergence is in vector fields?

Correct me, please, if I'm misusing the terminology.

Just curious...

If there is no such relationship, as is probably the case, could someone tell me what the purpose of the Laplacian, in multivariate functions,

*is*exactly?Or, more generally, what the purpose of divergence is in vector fields?

Correct me, please, if I'm misusing the terminology.

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