Lorentz transform $\neq$ Jacobi matrix?

Question

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 $$da'^\mu=\frac{\partial x'^\mu}{\partial x^\lambda} da^\lambda\mathrm{~?}$$

Answer 1

Actually Lorentz transform = Jacobi matrix. This is because Lorentz transformations are linear. Probably your textbook is going to discuss nonlinear transformations, like the ones encountered in general relativity. In that case, the equation you wrote is the transformation law of the coordinates of a tangent vector.