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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

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As a linear transformation (it may actually be an affine mapping, but the translation part does not change things here) a reflection typically has a single eigenvalue $-1$ and several $+1$:s. So on a line I would expect to see $-1$. In higher dimensions a matrix with appropriate eigenvalues. In an infinite dimensional space...? –  Jyrki Lahtonen May 22 at 6:17

1 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 $A$).

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