doesn't this feels overly familiar to magnetic induction and the current direction...?
coincidence?

dchen2

This seems to say that in 2D, there is one (essentially unique) notion of change for vector fields. In 3D, we had div and curl. So in some n-dimensional space, would there be n - 1 notions of change?

David

I think it's not a coincidence how it is similar to magnetic induction. The magnetic induction is kind of like a vector field that allows you to see the direction and magnitude of the magnetic force at given points

birb

Is there an intuitive explanation for this relationship between curl and divergence?

embl

Would this relationship be relevant for graphics applications?

anag

Are there any more notions of change besides divergence and curl of vector fields in higher dimensions?

coolpotato

Intuitively, if the divergence indicates how much a field is changing in size, curl represents how much a field is spinning, how does the divergence of X equal the curl of X (after rotating it 90 degrees)? Since we are working with 2-dimension fields, it is hard to geometrically picture the divergence and curl. Could you elaborate on what divergence and curl represent in a 3-dimensional space?

dl123

Could you provide more examples of fields that would use this property?

rbm

These ideas can sometimes be useful for sailing (kind-of 3d!)

doesn't this feels overly familiar to magnetic induction and the current direction...? coincidence?

This seems to say that in 2D, there is one (essentially unique) notion of change for vector fields. In 3D, we had div and curl. So in some n-dimensional space, would there be n - 1 notions of change?

I think it's not a coincidence how it is similar to magnetic induction. The magnetic induction is kind of like a vector field that allows you to see the direction and magnitude of the magnetic force at given points

Is there an intuitive explanation for this relationship between curl and divergence?

Would this relationship be relevant for graphics applications?

Are there any more notions of change besides divergence and curl of vector fields in higher dimensions?

Intuitively, if the divergence indicates how much a field is changing in size, curl represents how much a field is spinning, how does the divergence of X equal the curl of X (after rotating it 90 degrees)? Since we are working with 2-dimension fields, it is hard to geometrically picture the divergence and curl. Could you elaborate on what divergence and curl represent in a 3-dimensional space?

Could you provide more examples of fields that would use this property?

These ideas can sometimes be useful for sailing (kind-of 3d!)

How curl does 90 degree rotation in more than 2d?