The Big Theorems Live in 3D
In L6 you saw the Gauss and Stokes theorems as circles in a flat plane. But that was a flattened shadow of the real thing. The true stage is three dimensions. The divergence theorem is about a sphere in space, not a circle in a plane, and its boundary is a surface, not a rim. The Stokes theorem holds through any surface you cap a loop with, not just a flat disk. In this lesson you rotate the view with your own finger to see, in solid form, how a normal stands up on a surface, how flux passes through a sheet, and how the sum inside comes out equal to the boundary. The two lines you memorized in the plane come back to life over volumes and surfaces.
First see, in solid form, what a surface even is. Floating on the screen is a saddle sheet, z = 0.4(x² − y²): it rises one way and falls the perpendicular way, like a horse's saddle. Drag the screen with your finger to rotate it. What looked like a flat picture turns out to be a two-dimensional sheet hanging in space. At one point on the surface a normal arrow n̂ stands straight up, in the direction exactly perpendicular to the sheet right there. Bend the saddle harder with the slider and the normal tilts along with it. A surface is a sheet, and you can stand one normal anywhere on it: that is where surface integrals begin.
Now place that surface in a flow. There is a field streaming evenly upward, F = [0, 0, 1.4], and a flat square sheet floating in that flow, studded with outward normal arrows. The surface flux ∫∫ F·n̂ dS is the total amount of flow passing out through this sheet: on each little patch you take F·n̂ times the patch's area and add them all up. Move the tilt slider to turn the sheet to face the flow, then stand it on edge. Facing the flow head on, the flux is largest; standing sideways, parallel to the flow, it drops to nearly zero. Drag to rotate the view and the angle between the normal and the flow shows in solid form. Flux is, in the end, how much the surface faces the flow.
Now the real stage of the divergence theorem. The circle in L6 was really a sphere's shadow. The field is F = [x, y, z], a flow streaming outward from the origin in all directions, with divergence ∇·F = 3 everywhere. A translucent sphere floats on the screen, ringed with outward normal arrows on its surface. Grow and shrink the sphere with the radius slider. Two numbers follow along. Boundary is the total flux leaking out through the sphere's surface, ∮∮ F·n̂ dS; inside is the volume integral that adds up all the divergence in the ball, ∭ ∇·F dV. At any radius the two line up almost exactly, because both equal 4πR³. Drag to rotate the view and even the arrows on the far side come into view. It is the same theorem as the plane's, with the rim raised to a surface and the area raised to a volume, one dimension up.
The Stokes theorem shows its real power in 3D. The field is F = [−y, x, 0], a vortex spinning counterclockwise in the plane, with curl ∇×F = (0, 0, 2). On the screen a closed circular loop sits in the z=0 plane, with tangent arrows running around the rim. Press the toggle. It swaps the capping surface you stretch over this loop between a flat disk and a dome bulging upward (a hemisphere). Here is the point: the two have completely different shapes, yet they share the same loop as boundary. Boundary is the circulation around the loop, ∮ F·dr; surface is the curl flux through the cap, ∫∫ (∇×F)·n̂ dS; the two line up almost exactly, and whether you make the cap a disk or a dome that value stays the same. Grow the loop with the radius slider and drag to rotate the view. The circulation is fixed by the boundary alone, and the curl sums to the same thing through any surface that seals that boundary.
Last, overlay 2D and 3D on one screen. The toggle moves between two modes. Turn on 3D and a sphere ringed with outward normals (Gauss) or a loop with a cap (Stokes) floats in solid form. Turn on 2D and it flattens into the very picture you saw in L6: a circle in the plane ringed with outward arrows for Gauss, or with tangent arrows running around it for Stokes. An inner toggle swaps Gauss and Stokes too. Drag to rotate the view and watch the solid sphere and loop tip onto their side and collapse into the flat picture. The 2D theorems of L6 were not wrong; they were this 3D theorem seen as a shadow, looking down the z-axis. The two lines ∮∮ F·n̂ dS = ∭ ∇·F dV and ∮ F·dr = ∫∫ (∇×F)·n̂ dS close the whole of vector calculus.