Ok so. There are, in essence, two ways to define the two derivatives of a vector field, curl and divergence. The first way is to define a differentiation operator, del, or ∇:
∇ = (d/dx, d/dy, d/dz)
As ∇ is a pseudovector, there are two ways to multiply it by a vector field, V.
The first is the scalar, or dot product, which gives the divergence:
∇.V = div(V)
The second is the vector, or cross product, which gives the curl:
∇ x V = curl(V)
There is also a different way to define divergence and curl.
Divergence at a point is the ratio of the flux (surface integral of field out of a closed surface) over an infinitesimal surface enclosing an infinitesimal volume to that volume itself:
div = Φ/V
Curl is the infinitesimal circulation around a closed loop enclosing a point (I'm a bit fuzzier on the exact definition of curl).
If you extend these 'geometric' definitions of what the divergence and curl of a field V represent, you get two higher dimensional versions of the fundamental theorem of calculus, Stokes' theorem:
Surface integral of curl = line integral of field
And the divergence theorem:
Volume integral of divergence = surface integral of field
So my question is:
How can these two definitions be interpreted together?
What does Stokes' theorem/the divergence theorem suggest about the respective 'dimensions' of curl and divergence - clearly the disparity in the 3D/2D/1D integrals means something - and how does this relate to the del definition?
I'm not looking for a mathematical proof that these two definitions give equivalent answers, I know the proof. I'm asking for a intuitive reasoning as to why the 'scalar product derivative' of a vector field should have anything to do with volume and why the 'vector product derivative' should have anything to do with surface.