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# Stokes' theorem intuition

Conceptual understanding of why the curl of a vector field along a surface would relate to the line integral around the surface's boundary. Created by Sal Khan.

## Want to join the conversation?

• I don't understand why we have to dot the curl of F with the normal vector. Since we are integrating over the surface aren't we integrating F on the surface anyway? •  Stokes theorem says that ∫F·dr = ∬curl(F)·n ds.
If you think about fluid in 3D space, it could be swirling in any direction, the curl(F) is a vector that points in the direction of the AXIS OF ROTATION of the swirling fluid. curl(F)·n picks out the curl who's axis of rotation is normal/perpendicular to the surface. So if the axis of rotation is perpendicular to the surface, then the swirling is happening parallel to (and right on) the surface.
• I still don't really understand Stokes theorem. Can someone help explain it to me? •  I think this is the best way to teach it.

It is a speccial case of the Holographic principle...The Holographic Principle means that information inside a reigon is completely encoded onto its boundary.

So, in Stokes' theoremn, the flux through a reigon (the surface \$S\$ ) is exactly the same as the work through its boundary (the curve \$C\$).
• What would happen if we had a circular surface where at each point on the boundary the vector was perpendicular. Would that have curl or would it be equivalent to the 2nd example Sal gives where it all cancels out? (Or is this in fact not at all related?) • If the all of the vectors on the surface were perpendicular (or normal) to the surface, then the curl of the vector field would always be zero along the surface, so the surface integral of the curl is 0. Also, for the line integral, the dot product is 0 because they are perpendicular, so the line integral is 0.
So yes, both the surface integral of the curl and the the line integral would be zero.
• How is n ds the same as d s ? Also ds is supposed to be a differential of area? • Which surface are we talking of exactly? There are tons of surfaces which can have the boundary C right? Are we talking of any of those surfaces? • In principle, given a boundary C, Stokes theorem also holds for any 3-D surface with boundary C. Roughly speaking, this means that with any closed curve C, along with any surface S with boundary C, the line integral of F around C is equal to the sum of the "curls" of F on the surface S. Sal started of with simple surfaces in this video to help the viewer develop an intuitive understanding of what Stokes theorem means physically.
• At ; by dotting the vector field with the normal wouldn't you get the projection of the vector field perpendicular to the surface? I understand that we need the projection of the vectorfield ON the surface, but I don't see how dotting with the normal vector accomplishes that? • Stokes theorem says that ∫F·dr = ∬curl(F)·n ds. We don't dot the field F with the normal vector, we dot the curl(F) with the normal vector.
If you think about fluid in 3D space, it could be swirling in any direction, the curl(F) is a vector that points in the direction of the AXIS OF ROTATION of the swirling fluid. curl(F)·n picks out the curl who's axis of rotation is normal/perpendicular to the surface.
• is there a video where curl is explained? • who was stokes? • Sir George Gabriel Stokes was an Irish physicist and mathematician (13 August 1819 – 1 February 1903) who made great contributions to fluid dynamics and optics.

Fun fact:
If you watch The Walking Dead and have watched season 5 or 6, you will know that there is a pastor named Gabriel Stokes.
Sir George Stokes, the creator of Stokes theorem, had a father who's name was Gabriel Stokes who also happened to be a pastor.

For all you Walking Dead fans, I'd like you to take a moment to appreciate this coincidence
• Around , does it matter whether we say that we calculate a double integral over a "surface" (symbol S) over over a "region" (symbol R)?  