Let $D(\mathbb{R}^n)$ be the space of test functions on $\mathbb{R}^n$. My question is whether the various kinds of integrals in multivariable calculus (such as line integrals, surface integrals, etc.) can be thought of as elements of $D'(\mathbb{R}^n)$ the space of distributions.
For example, given an oriented $C^1$ regular curve $\gamma:[0,1]\to\mathbb{R}^n$ we can define the line integral of of a test function $\varphi\in D(\mathbb{R}^n)$ as $$ u(\varphi)=\int_\gamma \varphi \, ds= \int_0^1 \varphi(\gamma(t)) \|\gamma'(t)\|\, dt $$ This map is clearly a linear functional, however, is it a continuous linear functional (i.e. a distribution)? Though I suspect it is, I'm not exactly sure how to show it.
Are surface integrals and their higher dimensional analogues also distributions? Does anything change if we replace $D(\mathbb{R}^n)$ with $S(\mathbb{R}^n)$ (the space of Schwartz functions)?
(I started wondering about this because in physics one often solves for a magnetic field in the presence of an infinitely thin current distribution. This has the feeling of a Dirac delta, but is not quite the same. I'm hoping for a distribution theory interpretation. If there is a superior theory for this kind of analysis I would also be interested.)
Sure it is continuous if $\gamma$ isn't too badly behaved. The definition of continuity is : for every compact $\Omega$ there exists $k_\Omega$ and $C_\Omega$ such that $$ \forall \varphi \in C^\infty_c(\Omega), \qquad |u(\varphi)| \le C_\Omega\sum_{m=0}^{k_\Omega} \sup_x |\partial^m \varphi(x)|$$
(in $\mathbb{R}^n$ we replace $\partial^m$ by $\partial^{\alpha} = \prod_{i=1}^n \partial_{x_i}^{\alpha_i}$, $\sum_{i=1}^n \alpha_i = m$)
Do you see what would be $k_\Omega,C_\Omega$ in this case ?