🧄Finding the Covariant Components of a Magnetic Field

In non-Cartesian coordinate systems, the basis vectors are not of unit length, causing the components to behave differently. The process uses the metric tensor as a conversion tool, which accounts for the varying scale of the coordinate system. The result shows that while the initial contravariant component $\left(B^\phi\right)$ depends on the radial distance $(\rho)$ from the wire, the final covariant component ( $B_\phi$ ) is constant, directly representing the physical field strength after accounting for the basis vector's length.

Finding the Covariant Components of a Magnetic Field | Proof and Derivationviadean on Notion