User:Gglanzani/Gaussian Integral

The Gauss integral has numerous applications in geometry. One of this is the computation of the linking number a numerical invariant that describes the linking of two closed curves in three-dimensional space.

Definition
Let $$M$$ be a differential manifold, compact and oriented, $$n$$-dimensional. Let $$\phi: M \rightarrow S^n$$ be a differentiable map, d$$O_n$$ the pull-back of a volume element of $$S^n$$ under $$\phi$$, and $$O_n$$ the volume of $$S^n$$. Then the Gauss integral of $$\phi$$ is

$$\frac{1}{O^n}\int_M \mathrm{d}O_n.$$

In case $$M$$ has no boundary, the integral gives the degree of $$\phi$$.