Abstract:

Topology has been widely discussed in quantum mechanics in recent years. In this paper，the integral

formula of Ampère s circuital law is re-expressed as the integral formula of the first Chern number of a vector

field defined on the torus parametric surface. The numerical simulation shows that the integral value is an integer，i.

e.，the first Chen number，which represents the global property of vector field: When the vector field undergoes

continuous transformation，the local value of the vector field changes，however the integral value，i.e.，the Chern

number，remains unchanged; if the Chern number changes，it implies the vector field has experienced a discontinuous

change and there rises the singularity in the distribution of vector field. Furthermore，the vector field is

mapped from torus parametric surface to the unit sphere by Gauss mapping，and the intuitive geometric meaning of

the first Chern number is given. The theoretical and numerical results reveal the topological nature of Ampère s circuital

law，and show that the concept of topology can also be widely applied in classical physics.

Key words: Ampère s circuital law, topology, Chern number