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For an oriented smooth surface and compatible boundary orientation, apply Stokes theorem to $\varphi \mathbf{k}$ for an arbitrary constant vector $\mathbf{k}$. Since $\nabla \times (\varphi \mathbf{k})=\nabla \varphi \times \mathbf{k}$, its surface integral is $\mathbf{k}\cdot {\int }_{S}d\mathbf{S}\times \nabla \varphi$. The boundary integral is $\mathbf{k}\cdot \oint \varphi d\mathbf{x}$. Equality for every $\mathbf{k}$ proves the displayed vector identity. Reversing both orientations preserves the identity.

For sufficiently smooth scalar fields near an oriented surface, the product rule for the curl gives $\nabla \times (\varphi \nabla \psi )=\varphi \nabla \times \nabla \psi +\nabla \varphi \times \nabla \psi =\nabla \varphi \times \nabla \psi$. Stokes theorem then proves the displayed identity with consistent boundary orientation. A scalar multiple of a gradient need not itself be a gradient; the cross product of the two gradients detects its local circulation.

A charged brane couples to an appropriate differential-form gauge potential through its pullback. Under $A\mapsto A+d\Lambda$, a closed worldvolume changes the action by a boundary term, by Stokes theorem. Reversing the brane charge reverses this coupling, which matters in probe-force and supersymmetry calculations.