# The theorem of the day, Stokes' theorem relates the surface integral to a line integral. Since we will be working in three dimensions, we need to discus what it means for a curve to be oriented positively. Let S be a oriented surface with unit normal vector N and let C be the boundary of S.

Stokes' theorem is the 3D version of Green's theorem. It relates the surface integral of the curl of a vector field with the line integral of that same vector field around the boundary of the surface:

Review of Curves. Intuitively, we think of a curve as a path traced by a moving particle in. Oct 29, 2008 line integral around the boundary of that surface. Stokes' Theorem can be used to derive several main equations in physics including the May 3, 2018 Stokes' theorem relates the integral of a vector field around the boundary ∂S of a surface to a vector surface integral over the surface.

Stokes’ theorem relates the flux integral over the surface to a line integral around the boundary of the surface. Note that the orientation of the curve is positive. Suppose surface S is a flat region in the xy -plane with upward orientation. This classical Kelvin–Stokes theorem relates the surface integral of the curl of a vector field F over a surface (that is, the flux of curl F) in Euclidean three-space to the line integral of the vector field over its boundary (also known as the loop integral). Simple classical vector analysis example Stokes' Theorem relates line integrals of vector fields to surface integrals of vector fields.

## Key topics include vectors and vector fields, line integrals, regular k-surfaces, flux of a vector field, orientation of a surface, differential forms, Stokes' theorem,

Stokes’ Theorem Let S S be an oriented smooth surface that is bounded by a simple, closed, smooth boundary curve C C with positive orientation. Also let →F F → be a vector field then, ∫ C →F ⋅ d→r = ∬ S curl →F ⋅ d→S ∫ C F → ⋅ d r → = ∬ S curl F → ⋅ d S → The classical Stokes' theorem can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface. Stokes' theorem is a special case of the generalized Stokes' theorem.

### The theorem of the day, Stokes' theorem relates the surface integral to a line integral. Since we will be working in three dimensions, we need to discus what it means for a curve to be oriented positively. Let S be a oriented surface with unit normal vector N and let C be the boundary of S.

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In fact, Stokes' Theorem provides insight (∇ × F) · dS for each of the following oriented surfaces S. (a) S is the unit sphere oriented by the outward pointing normal. (b) S is the unit sphere oriented by the Gauss' Theorem enables an integral taken over a volume to be replaced by one taken over the surface bounding that volume, and vice versa. Why would we want Surfaces Orientation = direction of normal vector field n. If a curve is the boundary of a surface then the orientations of both can be made to be compatible.

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### To use Stokes’ Theorem, we need to rst nd the boundary Cof Sand gure out how it should be oriented. The boundary is where x2+ y2+ z2= 25 and z= 4. Substituting z= 4 into the rst equation, we can also describe the boundary as where x2+ y2= 9 and z= 4. To gure out how Cshould be oriented, we rst need to understand the orientation of S.

(a) In a direct way (using the parameterization of the surface) (b) S is a closed surface ⇒ we can apply the Gauss theorem. 3 (b) using the Stokes' theorem.