# Lorentz transform $\neq$ Jacobi matrix?

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

• I don't know what the "$da$" are supposed to be in your second equation - what is $d$? Also, since a transformation between inertial frames is just a Lorentz transformation, $x' = \Lambda x$ and the Jacobian is the Lorentz transformation, I'm not sure what the problem is. – ACuriousMind Nov 20 '16 at 13:37
• By $da$ i mean an infinitesimal change of the vector components. Your last sentence confuses me. As far as I know the term Jacobian refers to the determinant of the Jacobi matrix. How can this be the lorentz transformation? – OD IUM Nov 20 '16 at 14:39