Taking the time derivative of a curl

[nabeel17]

Is the time derivative of a curl commutative? I think I may have answered this question... Only the partial time derivative of a curl is commutative? The total time derivative is not, since for example in cartesian coordinates, x,y,and z can themselves be functions of time. In spherical and cylindrical coordinates, even the unit vectors depend on time? Also the partial derivative of a curl in curvilinear coordinates is commutative?

 

[Khashishi]

I think you are confusing coordinate variables (which are usually called x, y, z, t) with position variables (which are also usually called x, y, z, t). The difference is that coordinate variables refer to a fixed grid over which a vector field is defined, and position variables refer to some dynamic position of some object.

A field has a value at each coordinate position, whereas an object only occupies a particular subset of coordinate positions. The time derivative and curl are commutative since they are operations on a field, and the field coordinates are (typically) orthogonal. It doesn't make sense to take the curl of an object position.