Derivative of rigid motion like reflection?

Question

Is it possible to define a derivative for rigid transformations eg. reflection and translation?

I am especially interested on reflections shortly $\sigma$.

Because I am trying to relate $\int_{A}f(\sigma x)dx$ with $\int_{\sigma A}f(x)dx$, where ,say, f is an integrable function $f:X\to [0,1]$ and $A\subset X$ Borel set.

So if we try change of variables, then we need $det(D \sigma)$. Feel free to go as advanced and technical as you like.

Does $lim_{h\to 0} \frac{\sigma(x+h)-\sigma(x)}{h}$ exist? I think it equals 1 because

$|x-y|=|\sigma(x)-\sigma(y)|\Rightarrow lim_{h\to 0} |\frac{\sigma(x+h)-\sigma(x)}{h}|= lim_{h\to 0} \frac{|x+h-x|}{h}=1$

Thanks

Answer 1

Rigid motions are just multiplications with a matrix (plus a translation), i.e. $x\mapsto Ax+b$. Then the derivative is simply $A$. (In fact, to be "rigid", i.e. an isometry, wwe require some conditions about $A$, but even without these conditions the derivative of $x\mapsto Ax+b$ is simply $A$).