Let M be a riemannian manifold and Mk is the space of k-dimensional compact sub-manifolds. Let f∈C∞(M) be a smooth function over M and X∈TMk a tangent vector of Mk at ˜M, X=(Xx)x∈˜M∈TM˜M. Define:
F(˜M)=∫˜Mf(x)dx
Then have we:
X(F)(˜M)=∫˜MXx(f)(x)dx
?