Let $f\in C^\infty(\mathbb{R}^n,\mathbb{R}^k)$, and $g\in C^\infty(\mathbb{R}^n,\mathbb{R})$, where $k<n$. How do I compute $$\int_{\mathbb{R}^n}\delta^k(f(\mathbf{x}))\cdot g(\mathbf{x})\mathrm{d}\mathbf{x}$$ where $\delta$ is the dirac-delta function? I have a hypothesis: $$\int_{\mathbb{R}^n}\delta^k(f(\mathbf{x}))\cdot g(\mathbf{x})\mathrm{d}\mathbf{x}=\int_{f^{-1}(\mathbf{0})}g(\mathbf{y})\mathrm{d}\mathbf{y}$$ Where $\mathrm{d}\mathbf{y}$ is the euclidean volume form on the manifold $~f^{-1}(\mathbf{0})\subset \mathbb{R}^n$.
Is this correct?