April 21, 2026
Some good news: our paper “Nonlocal vector calculus on the sphere,” joint with Mikaël Slevinsky and Qiang Du, has just been accepted for publication in M3AS. I’m particularly happy about this one—it’s a project where the mathematics turned out cleaner than expected, and the main idea is quite satisfying.
This post is a short attempt to explain what we did, and in particular what I think is the most interesting result: a nonlocal version of Stokes’ theorem on the sphere.
Classical vector calculus is local. Gradients, divergences, curls—they all depend on derivatives, and therefore on what happens infinitesimally close to a point. This works beautifully when functions are smooth and the geometry is simple.
But in many applications—materials science, peridynamics, data analysis—you don’t want locality. You want interactions over a finite distance. Points “talk” to each other across a neighborhood, not just through derivatives.
This leads to nonlocal operators: instead of derivatives, you integrate differences over a region. The price you pay is that the clean structure of vector calculus starts to break down. The gain is that you can model much richer phenomena.
In flat space, this nonlocal framework is by now relatively well understood, and analogues of classical identities—including Stokes-type theorems—have already been developed.
The question we ask here is slightly different: what happens when you try to do the same thing on a curved surface?
Extending nonlocal calculus from flat space to a curved surface like the sphere is not just a technical step—it fundamentally changes the problem.
On the sphere, everything is constrained to the tangent plane. Vectors at different points live in different spaces, and there is no canonical way of comparing them. Even defining something as simple as a “difference between two points” requires care.
This is where most of the difficulty lies. In flat space, geometry is invisible; on the sphere, it is everywhere.
What we do in the paper is to define nonlocal analogues of the classical operators—gradient, divergence, curl—using carefully constructed integral kernels that respect the geometry of the sphere.
The key idea is that these operators can still be analyzed using spherical harmonics. In fact, one of the main technical results is that they are diagonal in that basis. This makes the analysis much cleaner than one might expect.
The highlight of the paper, at least to me, is the following: we prove a nonlocal Stokes theorem on the sphere.
In the classical setting, Stokes’ theorem relates the circulation of a vector field along a boundary to the curl of the field inside the surface. It is one of the cornerstones of vector calculus.
Nonlocal versions of this theorem already exist in Euclidean space. The novelty here is to establish such a result on a curved surface. There is no a priori reason why the same structure should survive: curvature introduces additional geometric constraints.
What we show is that, with the right definitions, a nonlocal Stokes-type identity still holds on the sphere.
Let \(\Sigma \subset S^2\) be a spherical patch with boundary \(\partial \Sigma\), and let \(\boldsymbol{s}\) denote the unit tangent vector along \(\partial \Sigma\). In the classical setting, Stokes’ theorem reads
\[ \int_{\Sigma} (\nabla \times \boldsymbol{V}) \cdot \boldsymbol{x}\, d\Omega = \int_{\partial \Sigma} \boldsymbol{V}\cdot \boldsymbol{s}\, d\omega. \]
In our paper, the nonlocal operators are built from kernels adapted to the geometry of the sphere. We restrict attention to antisymmetric kernels of the form
\[ \boldsymbol{\beta}(\boldsymbol{x},\boldsymbol{y}) = \nabla_{\boldsymbol{x}} \gamma(\boldsymbol{x}\cdot \boldsymbol{y}) - \nabla_{\boldsymbol{y}} \gamma(\boldsymbol{x}\cdot \boldsymbol{y}) = \gamma'(\boldsymbol{x}\cdot \boldsymbol{y})\,(\boldsymbol{y}-\boldsymbol{x}), \]
where \(\gamma\) is a scalar kernel depending only on the dot product \(\boldsymbol{x}\cdot \boldsymbol{y}\).
To localize interactions to a finite horizon \(\delta>0\), we choose
\[ \omega_\delta(\boldsymbol{x},\boldsymbol{y})=\chi_{[0,\delta]}(|\boldsymbol{x}-\boldsymbol{y}|), \qquad \gamma_\delta(\boldsymbol{x}\cdot \boldsymbol{y}) = \gamma(\boldsymbol{x}\cdot \boldsymbol{y})\,\omega_\delta(\boldsymbol{x},\boldsymbol{y}), \]
and then set
\[ \boldsymbol{\beta}_\delta(\boldsymbol{x},\boldsymbol{y}) = \gamma_\delta'(\boldsymbol{x}\cdot \boldsymbol{y})\,(\boldsymbol{y}-\boldsymbol{x}). \]
With this notation, the weighted nonlocal curl of a vector field \(\boldsymbol{V}\) is
\[ C^\delta\{\boldsymbol{V}\}(\boldsymbol{x}) = \int_{S^2} \boldsymbol{\beta}_\delta(\boldsymbol{x},\boldsymbol{y}) \times \bigl[\boldsymbol{V}(\boldsymbol{y})-\boldsymbol{V}(\boldsymbol{x})\bigr]\, d\Omega(\boldsymbol{y}). \]
The key result is that this nonlocal curl satisfies a Stokes-type identity on the sphere. More precisely, if we define the averaging operator
\[ A^\delta\{\boldsymbol{V}\}(\boldsymbol{x}) = \int_{S^2} \gamma_\delta(\boldsymbol{x}\cdot \boldsymbol{y})\, \boldsymbol{V}(\boldsymbol{y})\, d\Omega(\boldsymbol{y}), \]
then one proves that
\[ \int_{\Sigma} C^\delta\{\boldsymbol{V}\}\cdot \boldsymbol{x}\,d\Omega = \int_{\partial\Sigma} A^\delta\{\boldsymbol{V}\}\cdot \boldsymbol{s}\,d\omega. \]
This can be seen as the curved analogue of nonlocal Stokes formulas in flat space. The structure is very similar, but with an important difference: the boundary term involves the averaged field \(A^\delta\{\boldsymbol{V}\}\), reflecting the nonlocal nature of the model.
The key mechanism behind this result is that the nonlocal operators are diagonal in the basis of vector spherical harmonics. This makes it possible to prove the identity
\[ C^\delta\{\boldsymbol{V}\}\cdot \boldsymbol{x} = (\nabla\times A^\delta\{\boldsymbol{V}\})\cdot \boldsymbol{x}, \]
and once this is established, the result follows by applying the usual local Stokes theorem to the averaged field.
Finally, as \(\delta\to 0\), the interaction range shrinks, the averaging disappears, and the nonlocal operators converge to the classical ones. In that limit, we recover the usual Stokes theorem on the sphere.
This project was a nice mix of geometry, analysis, and spectral methods. What I find most interesting is that the main difficulty is not the nonlocality itself—this is by now well understood in flat space—but the interaction between nonlocality and curvature.
Many classical structures in mathematics are surprisingly robust. But they are not automatic: on a curved space, you have to work to make them reappear.
And of course, it’s always nice when a paper finds a home. Thanks to my coauthors for a great collaboration, and to the editors for handling the submission.