Stokes’ theorem

In Stokes' theorem, $S$ denotes the specific chosen oriented surface whose boundary is $C$. The base-plane projection is merely a visual aid to help visualize the geometry and does not automatically represent the surface $S$ itself.

Conditions: The visualization includes a shaded base region.; The theorem refers to an oriented surface $S$ with boundary $C$.

The boundary orientation must follow the right-hand rule to ensure consistency between the direction of traversal along the curve $C$ and the chosen normal vector of the surface $S$. If the thumb of the right hand points in the direction of the normal vector, the fingers curl in the positive direction of traversal for the boundary curve.

Conditions: The surface $S$ has a chosen oriented normal vector.; The boundary curve $C$ is traversed in a specific direction.

Stokes' formula equates the line integral of a vector field $\vec{F}$ around the closed boundary curve $C$ with the surface integral of the curl of $\vec{F}$ dotted with the oriented unit normal over the surface $S$. The left side accumulates tangential field components along the boundary, while the right side accumulates normal components of the curl across the surface.

Conditions: The surface $S$ is an oriented piecewise smooth surface.; The vector field $\vec{F}$ is $C^1$ near the surface.; The boundary orientation of $C$ is induced by the chosen normal of $S$ via the right-hand rule.

A single upward normal arrow is insufficient because normals generally vary across a curved surface. The orientation of the surface is defined by a continuous field of normal vectors, not a constant direction.

Conditions: The surface $S$ is curved (not planar).; The theorem requires an oriented surface with a consistent normal field.