# Question regarding space transformation in special relativity

While learning about Special Relativity I learnt that we use the Transformation matrix to alter the space .This matrix differs for $$Contravariant$$ and $$Covariant$$ vectors.Why does it happen?,Why one kind of matrix ($$Jacobian$$) for $$basis$$ $$vectors$$ and other kind($$Inverse$$ $$Jacobian$$) for $$gradient$$ ,$$divergence$$ etc.Why does the matrix change ,why does changing quantities require a new matrix and why does $$Inverse$$ $$Jacobian$$ satisfy this?

$$My$$ $$understanding$$ $$of$$ $$how$$ $$Jacobian$$ $$achieves$$ $$this$$:

https://youtu.be/bohL918kXQk

Here Grant says that by moving a small dist $$\partial x$$ ,we produce a change in $$\partial f_1$$ direction and $$\partial f_2$$ direction , $$\partial f_1$$ in being $$x+\sin(y)$$ direction & $$\partial f_2$$ being in $$y+\sin(x)$$ ,(ie) $$\hat{x}$$ transforms to $$x+\sin(y)$$ and $$\hat{y}$$ axis transforms to $$y+\sin(x)$$, by moving a small distance $$\partial x$$ in $$x$$ axis in input we produce a effect on both the axes in output space, similar thing happens in y direction .

$$(x,y) \rightarrow (x+\sin(y),y+\sin(x))$$  $\partial x$ in x direction in input space (input direction) causes a change in both the axes in output space" />

$$J = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \\ \end{bmatrix}$$

Is my interpretation a right one?

And why does the $$Jacobian$$ Matrix change for covariant and contravariant vectors?

For a small distance, the coordinates $$\left(\begin{matrix}\hat x\\ \hat{y}\end{matrix}\right)=\left(\begin{matrix}f_1(x,y)\\ f_2(x,y)\end{matrix}\right)=:f(x,y)$$ change by $$\left(\begin{matrix}\Delta \hat x\\ \Delta \hat y\end{matrix}\right)=J\cdot \left(\begin{matrix}\Delta x\\ \Delta y\end{matrix}\right)\,.$$ That's the transformation law for contravariant vectors. Conversely, by the chain rule, \begin{align} \partial_{x}(F\circ f)&= ((\partial_{\hat x}F)\circ f)\,\partial_x f_1+((\partial_{\hat y}F)\circ f)\,\partial_x f_2\\[3mm] \partial_{y}(F\circ f)&= ((\partial_{\hat x}F)\circ f)\,\partial_y f_1+((\partial_{\hat y}F)\circ f)\,\partial_y f_2\,. \end{align} This can be written as $$\nabla (F\circ f)=J^\top\cdot ((\hat\nabla F)\circ f)$$ or as $$\nabla =J^\top\cdot\hat\nabla\,.$$ This is the transformation law for covariant vectors (gradients).