Whilw trying to understand Maxwell's equations (having learned a bit about manifolds) I encounter the following issue:

Gauss' Law seems to integrate the electric field $E$ over a $2$-manifold, seemingly implying that $E$ is a $2$-form.

Ampere's Law of Induction, however, integrates the electric field over a $1$-manifold, seemingly implying that $E$ is a $1$-form.

Which one is it? Is $E$ a $1$-form or a $2$-form? And if $E$ is an $m$-form, how is integration carried out on a $n$-manifold for $m\neq n$?

While trying to understand Maxwell's equations (having learned a bit about manifolds) I encounter the following issue:

Gauss' Law seems to integrate the electric field $E$ over a $2$-manifold, seemingly implying that $E$ is a $2$-form.

Ampere's Law of Induction, however, integrates the electric field over a $1$-manifold, seemingly implying that $E$ is a $1$-form.

Which one is it? Is $E$ a $1$-form or a $2$-form? And if $E$ is an $m$-form, how is integration carried out on a $n$-manifold for $m\neq n$?