Let's assume the vector field is NOT a gradient field.
Are there any restrictions on what the curl of this vector field can be?
If so, how can I determine a given curl of a vector field can NEVER be a particular vector function?
Here's something interesting about this problem.
Does it take more work to pump water from bottom of tank or from top of tank as you slowly lower the hose?
How do I derive this vector integral from the simple divergence theorem?
I seem to lose the vector if I start off with
Scalar function times arbitrary constant Vector V as my starting vector field.
The surface integral
Scalar function times vector dS
Does NOT make sense.
Furthermore, Volume integral of gradient of the scalar function times dV makes no sense either.
Equating these two integrals to each other just does not produce meaning as there is clear meaning of the divergence...
The divergence theorem states that the flux of the vector field through the surface is equal to the divergence of the vector field throughout the volume. So, no I do not have the same problem with the divergence theorem