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Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations: \begin{equation} f_1(\vec x)=0, \\ \vdots \\ f_m(\vec x)=0, \end{equation} (where $\vec x$ are ...

Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of polynomial equations: \begin{equation} P_1(\vec x)=0, \\ \vdots \\ P_m(\vec x)=0, \end{equation} (where $\...