Application of the fundamental theorem of calculus in $n$ dimensions: Stokes', divergence or gradient theorem?

Question

Consider the following linear integral solved by the Fundamental Theorem of Calculus: $$\int_{a_1}^{a_2}dx\, e^{ixk}=\int_{a_1}^{a_2}dx\, \frac{d}{dx}\left(\frac{e^{ixk}}{ik}\right)= \frac{e^{ixk}}{ik}\Biggr\rvert_{a_1}^{a_2}=\frac{e^{ia_2 k}- e^{i a_1 k}}{ik}\,.$$ Is there a way to make use of a generalization of this theorem in $n$ dimensions (maybe the Gradient Theorem, or Divergence Theorem, or Stokes' theorem?) such that the multidimensional integral defined over a $n$-dimensional generic volume $V$ $$\int_{V}d\mathbf{x}\, e^{i\mathbf{x}\cdot \mathbf{k}}$$ is solved in terms of the integrand function evalutated at the $(n-1)$-dimensional boundary surface $\partial V$ of the volume $V$: $$\int_{V}d\mathbf{x}\, e^{i\mathbf{x}\cdot \mathbf{k}}=\int_V d\mathbf{x}\, \nabla_{\mathbf{x}}(e^{i\mathbf{x}\cdot \mathbf{k}})\cdot \frac{\mathbf{k}}{i k^2} \propto e^{i\mathbf{x}\cdot \mathbf{k}}\rvert_{\mathbf{x}\in \partial V}\, ?$$ Indeed in the linear example above, the integral solution is expressed in terms of the exponential function appearing in the argument, evaluated at the two points corresponding to the borders $x=a_1, a_2$ of the linear domain $[a_1,a_2]$.

Answer 1

What you want is known as the (Generalized) Stokes Theorem. It is stated in the language of differential forms.

A good introductory treatment can be found in Spivak's little book Calculus on Manifolds.

Very briefly, the integrands of classical vector calculus in 3-space, such as $dA \cdot \vec n$ for surface integrals or $dV$ for volume integrals, generalize to differential 2-forms and 3-forms respectively, and then to differential $k$-forms for any dimension $k \ge 1$. When you integrate over a "$k$-dimensional generic volume" as you put it, or over a "$k$-dimensional oriented manifold" in the new language, the thing that you integrate is a differential $k$-form. The differential operators of classical vector calculus, i.e. div, grad, curl and all that, generalize to an operator called the exterior derivative, or sometimes just the "differential", denoted $d$, which (for $k \ge 0$) inputs a differential $k$-form $\omega$ and outputs a differential $k+1$ form denoted $d\omega$.

Once all those prerequisites are developed (which takes most of Spivak's book), the generalized Stokes theorem then has a very simple formulation: $$\int_M d \, \omega = \int_{\partial M} \omega $$ where $M$ is a compact, oriented, $k$-dimensional manifold, $\partial M$ is its boundary which is a compact, oriented $k-1$-dimensional manifold, and $\omega$ is a $k-1$ form.