You may have noticed that the four-dimensional gradient operator /x functions like a covariant 4-vector--in fact, it is often written , for short. For instance,

You may have noticed that the four-dimensional gradient operator ∂/∂xμ functions like a covariant 4-vector--in fact, it is often written ∂μ, for short. For instance, the continuity equation, ∂μ Jμ = 0, has the form of an invariant product of two vectors. The corresponding contravariant gradient would be ∂μ ≡ ∂xμ. Prove that ∂μ Ф is a (contravariant) 4-vector, if Ф is a scalar function, by working out its transformation law, using the chain rule.