Curl and the Cross Product

  • #1
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?
 

Answers and Replies

  • #2
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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.
 

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