Divergence of a Magnetic Field not equaling zero

[Maliska]

I have a larger problem involving divergences and curls, but the correct answer requires ∇°B (divergence of B) = 0. I understand the proof of this in Griffiths, but the definition of divergence in cylindrical coordinates is:

After using the product rule to split the first term, we get the divergence of B is B_rho / rho + 0 + 0 + 0, or simply Div(B)=B_rho / rho; however, this clearly contradicts what we know about the divergence of B being zero. Can someone please clarify this for me, I've been stuck at it for hours.

Thanks.

P.s. First post!

 

[haruspex]

How do you know that the partial derivatives you claim to be zero are zero? Do you have a specific field in mind? A field that's a constant vector in Cartesian can look rather different in polar.

 

[Maliska]

Thank you haruspex for responding. The partial derivatives I said equal zero could be my mistake, but please explain how a vector with constant components in cartesian coordinates can have nonzero partial derivatives in cylindrical coordinates.

 

[haruspex]

A_ρ, for example, refers to the component of the vector in the ρ direction. If the vector is constant, its component in the ρ direction may yet depend on θ and z. It's not that the vector varies, but that the unit ρ-direction vector does.