# Derivation of functions over an infinite dimensional space

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Let $M$ be a riemannian manifold and $M_k$ is the space of $k$-dimensional compact sub-manifolds. Let $f\in{\cal C}^{\infty}(M)$ be a smooth function over $M$ and $X\in TM_k$ a tangent vector of $M_k$ at $\tilde M$, $X=(X_x)_{x\in \tilde M} \in TM^{\tilde M}$. Define:

$$F(\tilde M)=\int_{\tilde M} f(x)dx$$

Then have we:

$$X(F)(\tilde M)=\int_{\tilde M} X_x(f)(x) dx$$

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