Does Zero Volume Indicate Zero Curl in Vector Fields?

[rockyshephear]

Regarding the equation for curl:

Nable E literally means the sum of the differences of certain rates of change with respect to certain coordinates i hat, j hat, k hat.

Since Nabla Cross E also is interpreted as the volume of a paralleliped in 3D space...

1. when the volume is zero, does this mean there is zero curl?

2. when the volume is >0 to infinity, does that mean the rotation is happening faster or just dispersing faster thru the field or liquid or what have you?

Thanks,
Also, anyone know any good vector calculus jokes?

 

[fresh_42]

Curl refers to vector fields and volume of parallelepipeds to (normed) vector spaces. You cannot treat them as equal. Especially as volume requires a determinant and curl is another vector field. But if you take the partial derivatives as basis vectors and the vectors at a certain point as sides of your parallelepiped, then you get a volume element.