  Inspired by a recent paper of G. Liu and X. Tan (2023), we would like to
measure how the magnetic effect appears in the heat trace formula associated
with the magnetic Laplacian and the magnetic Dirichlet-to-Neumann operator. We
propose to the reader an overview of magnetic heat trace formulas through
explicit examples. On the way we obtain new formulas and in particular we
calculate explicitely some non local terms and logarithmic terms appearing in
the Steklov heat trace asymptotics.
