Contravariant vector ($T^{\prime a}=(\partial x^{\prime a}/\partial x^b)T^b$)

A contravariant vector has coordinate components obeying the displayed transformation law. A tangent vector to a parametrized curve provides the basic example: the chain rule transforms $d{x}^a/ds$ in precisely this way. Pairing it with a covector produces a coordinate-independent scalar. Conversely, if an array pairs invariantly with every contravariant vector, coefficient comparison forces the covector transformation law for that array.