Stokes’ Theorem Verification
Verify that Flux of Curl equals Boundary Circulation.
1. Vector Field F(x, y, z)
2. Surface S: z = g(x,y)
Domain Bounds (Rectangle)
∬ (Curl F) · dS
∮ F · dr
3D Visualization
Drag to RotateStokes’ Theorem: The swirl on the surface (Green/Blue) adds up to the flow around the red rim.
What is Stokes’ Theorem?
Stokes’ Theorem is the 3D “big brother” of Green’s Theorem. It relates a surface integral over a curved surface S to a line integral around its boundary curve C.
The Equation
∫∫S (∇ × F) · dS = ∮C F · dr
Sum of Curl on Surface = Circulation around the Rim
The “Hat” Analogy
Imagine a hat. The surface S is the fabric of the hat itself. The boundary C is the rim of the hat.
Stokes’ Theorem says that if you measure how much the air is swirling (Curl) at every tiny point on the hat’s fabric and add it all up, the result is exactly the same as measuring the air circulating just around the rim.
This is powerful because it means the shape of the hat (the surface) doesn’t matter! You can puff the hat up or flatten it down—as long as the rim stays the same, the total curl flux remains constant.
Frequently Asked Questions (FAQ)
Q: Why does the tool calculate two numbers?
A: To verify the theorem! We calculate the left side (Surface Integral) and the right side (Line Integral) separately using different math methods. If the theorem holds, the numbers should be nearly identical.
Q: What is “Curl”?
A: Curl (∇ × F) is a vector that measures the rotation or “swirliness” of a field at a point. Its direction is the axis of rotation.
Q: Does the direction of the boundary matter?
A: Yes. The boundary must be oriented consistent with the surface normal (Right-Hand Rule). If you walk along the rim with your head pointing in the normal direction, the surface should be to your left.