Question

Take a 3D surface R with boundary curve C. If you are given a vector field Field[x, y, z] with the extra property that curlField[x, y, z], with its tail at {x, y, z}, is tangent to the surface R at all points {x, y, z} on the surface R, then how does Stokes's formula tell you that the net flow of Field[x, y, z] along C is 0?

Answer 1

firstly,3D gradient field has zero curl

Stoke's Theorem says that the flow along is the double integral of CurlField[x,y,z] is unit normal.

Explanation:

3D gradient field has zero curl.

Stoke's Theorem says that the flow along is the double integral of CurlField[x,y,z] is unit normal. If CurlField[x,y,z] is tangent to the surface everywhere, it is perpendicular to the unit normal everywhere. So the flow is the integral of 0, and hence the net flow is 0.