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There is a formula that computes the integral of Hasdorff measures of level sets of a given function. It goes under the name "coarea formula". It is a generalization of a change of variables formula from multivariable calculus and of course the proof of the general formula proceeds by first showing it for linear functions. You can find all details for example in these lecture notes.

As you can see on page 13. the formula for linear mapping $A$ reads $$\int_{\mathbb{R}^m} \mathcal{H}^{m-n}(S \cap A^{-1}(y)) \mathrm{d}y = \sqrt{\mathrm{det}(LL^T)}\mathcal{L}^n(S),$$ where $\mathcal{H}^{m-n}$ denotes the $(m-n)$-dimensional Hausdorff measure, $\mathcal{L}^n$ denotes the $n$-dimensional Lebesgeue measure, $A^{-1}(y)$ denotes the set $\{x\in\mathbb{R}^n|Ax=y\}$ and $L$ is given by factoring the mapping $A$ as $A = LPQ$ where $Q$ is orthogonal, $P$ is projection onto the first $m$ coordinates and $L$ is a regular mapping $\mathbb{R}^d\to\mathbb{R}^d$.

There is a formula that computes the integral of Hasdorff measures of level sets of a given function. It goes under the name "coarea formula". It is a generalization of a change of variables formula from multivariable calculus and of course the proof of the general formula proceeds by first showing it for linear functions. You can find all details for example in these lecture notes.

As you can see on page 13. the formula for linear mapping $A$ reads $$\int_{\mathbb{R}^m} \mathcal{H}^{n-m}(S \cap A^{-1}(y)) \mathrm{d}y = \sqrt{\mathrm{det}(LL^T)}\mathcal{L}^n(S),$$ where $\mathcal{H}^{n-m}$ denotes the $(n-m)$-dimensional Hausdorff measure, $\mathcal{L}^n$ denotes the $n$-dimensional Lebesgeue measure, $A^{-1}(y)$ denotes the set $\{x\in\mathbb{R}^n|Ax=y\}$ and $L$ is given by factoring the mapping $A$ as $A = LPQ$ where $Q$ is orthogonal, $P$ is projection onto the first $m$ coordinates and $L$ is a regular mapping $\mathbb{R}^m\to\mathbb{R}^m$.

There is a formula that computes the integral of Hasdorff measures of level sets of a given function. It goes under the name "coarea formula". It is a generalization of a change of variables formula from multivariable calculus and of course the proof of the general formula proceeds by first showing it for linear functions. You can find all details for example in these lecture notes.

There is a formula that computes the integral of Hasdorff measures of level sets of a given function. It goes under the name "coarea formula". It is a generalization of a change of variables formula from multivariable calculus and of course the proof of the general formula proceeds by first showing it for linear functions. You can find all details for example in these lecture notes.

As you can see on page 13. the formula for linear mapping $A$ reads $$\int_{\mathbb{R}^m} \mathcal{H}^{m-n}(S \cap A^{-1}(y)) \mathrm{d}y = \sqrt{\mathrm{det}(LL^T)}\mathcal{L}^n(S),$$ where $\mathcal{H}^{m-n}$ denotes the $(m-n)$-dimensional Hausdorff measure, $\mathcal{L}^n$ denotes the $n$-dimensional Lebesgeue measure, $A^{-1}(y)$ denotes the set $\{x\in\mathbb{R}^n|Ax=y\}$ and $L$ is given by factoring the mapping $A$ as $A = LPQ$ where $Q$ is orthogonal, $P$ is projection onto the first $m$ coordinates and $L$ is a regular mapping $\mathbb{R}^d\to\mathbb{R}^d$.

Yes, there is and the general version of such formula for smooth (or Lipschitz, Sobolev, etc) mappings goes under the name "coarea formula". Basically, you need to compute the Hausdorff measure of the level sets of $f$ and integrate over all slices. And of course the proof of the general formula proceeds by first showing it for linear functions. You can find all details for example in these lecture notes.

There is a formula that computes the integral of Hasdorff measures of level sets of a given function. It goes under the name "coarea formula". It is a generalization of a change of variables formula from multivariable calculus and of course the proof of the general formula proceeds by first showing it for linear functions. You can find all details for example in these lecture notes.