A contravariant vector is an object that is usually written with a superscript and it is defined by the "transformation law":

$$V^{'i} = \frac{\partial x^{' i}}{\partial x^j} V^j $$

where $i,j = 0,1,2,3$.

In the definition above the fractional terms is referred to as the Jacobian. Now my confusion is the following: I understand that in the definition above really represent 4 different equations (due to the Einstein summation convention), but why does fractional term represent a matrix (the Jacobian).


You've probably seen other Jacobians in some class on multivariate calculus, when computing an integral and changing variables. The expression $\frac{\partial x'^i}{\partial x^j}$ has two indices, so we can regard it as a matrix $J^i_j$ that connects the primed and unprimed coordinate systems. If we write a position vector in these coordinates as

$$ [x^\mu]=\left(\begin{array}{c} x_1\\x_2\\ \vdots\\x_n \end{array}\right), \ \ \ \ [x'^\nu]=\left(\begin{array}{c} x'_1\\x'_2\\ \vdots\\x'_n \end{array}\right), $$ where $\mu,\nu=1,\cdots,n$, the transformation from one to another is a matrix $$ \left[\frac{\partial x'^\nu}{\partial x^\mu}\right]=\left(\begin{array}{cccc} \frac{\partial x'^1}{\partial x^1} & \frac{\partial x'^1}{\partial x^2} & \cdots & \frac{\partial x'^1}{\partial x^n}\\\frac{\partial x'^2}{\partial x^1} & \ddots\\ \vdots & & \ddots\\ \frac{\partial x'^n}{\partial x^1} & & &\frac{\partial x'^n}{\partial x^n} \end{array}\right), $$

so that $$ x'^\nu = \frac{\partial x'^\nu}{\partial x^\mu}x^\mu $$ in the sense of matrix multiplication (as you said, $n$ equations).


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