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I am currently learning about the theory of Normal Cycles which makes use of the language of currents and differential forms. They are defined in the following way

The Lipschitz-Killing curvature form of order $k \in\{0, \ldots, d-1\}$ is the $(d-1)$-form $\varphi_{k}$ on $\mathbb{R}^{2 d}$ given by $$ \begin{aligned} &\left\langle a_{1} \wedge \cdots \wedge a_{d-1}, \varphi_{k}(x, n)\right\rangle \\ &=\frac{1}{(d-k) \omega_{d-k}} \sum_{\substack{j_{1}+\cdots+j_{d-1}=d-k-1 \\ j_{i} \in\{0,1\}}}\left\langle\pi_{j_{1}}\left(a_{1}\right) \wedge \cdots \wedge \pi_{j_{d-1}}\left(a_{d-1}\right) \wedge n, \Omega_{d}\right\rangle, \end{aligned} $$ $a_{1}, \ldots, a_{d-1} \in \mathbb{R}^{2 d},(x, n) \in \mathbb{R}^{2 d}$. Since $\varphi_{k}(x, n)$ does not depend on the first vector coordinate $x$, we write usually only $\varphi_{k}(n)$. In particular, $$ \varphi_{0}=\left(d \omega_{d}\right)^{-1}\left(\pi_{1}\right)^{\#}\left(\mathrm{id} \sqsupset \Omega_{d}\right) . $$

with $\pi_0$ (resp $\pi_1$) are the projection on the first component (resp. the second) of $\mathbb{R}^d \times \mathbb{R}^d$, and $\Omega_d$ the classical volume form of $\mathbb{R}^d$.

I would like to compute their norm when taken at a point $(x,n)$ but I have neither found a reference in which it is done and I have yet to discover how to handle the computations.

Many thanks !

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