One of my textbooks says that a contravariant vector $a^\lambda$ transforms according to $$a'^\mu=\frac{\partial x'^\mu}{\partial x^\lambda} a^\lambda,$$ when changing the inertial frame of reference ($x^\mu\rightarrow x'^\mu$). At the same time $$a'^\mu=\Lambda_{\mu\lambda}a^\lambda,$$ where $\Lambda$ is the Lorentz transformation matrix.
This kind of confuses me as it implies the Jacobi-matrix w.r.t. to the basis vectors of the two frames of reference equals the Lorentz transformation. Shouldn't the first equation be $$da'^\mu=\frac{\partial x'^\mu}{\partial x^\lambda} da^\lambda\mathrm{~?}$$