Green Stokes Gauss theorems explained
Divergence measures how much leaks out of a point, and curl measures how much turns there. Sometimes you want the whole region, not one point. Add every little bit of divergence inside a region and the sum equals the total that leaves through the boundary. Add every bit of curl inside and the sum equals the circulation around the boundary. The first promise is Gauss's theorem. The second is Stokes's theorem. You can learn the total of what happens inside by measuring only the boundary. It is the shortcut that turns a wide area integral into a short integral around the rim. In each figure the disk is the region and the arrows on its rim are the measurement along the boundary.
Gauss's theorem, drawn as one circle. The background is a color map of divergence, gold where flow leaks out and blue where it is drawn in. Around the circle stand arrows pointing outward. Gold means flow leaving, and blue means flow entering. Grow or shrink the circle with the radius slider and two numbers follow together. Inside is the sum of the divergence in the disk. The boundary is the total flux crossing the rim. Whatever the radius, the two stay nearly equal. That is Gauss's theorem.
Now the picture steps aside and only two numbers remain, side by side. The left one is ∫∫ ∇·F dA, the area integral of every bit of divergence inside the circle. The right one is ∮ F·n ds, the flux leaking outward, gathered once around the rim. Move the radius slider and the two values travel together and stay nearly equal, and the match mark in the middle confirms it. One of them has to sweep the whole disk. The other only needs the boundary line. That is why the theorem is a shortcut.
This time it is Stokes's theorem, and the field has changed. One vortex turns counter-clockwise near the origin, and the background is a color map of curl. Gold is counter-clockwise turning, strongest at the center, and arrows along the circle point tangent to the rim. Move the radius slider and the inside number is the sum of the curl in the disk, while the boundary number is the circulation around the rim. Grow the circle and the two values grow together and keep pointing at the same number. The sum inside and the integral on the boundary agree, and the two readouts show it. The tangent arrows are the part of the flow that runs along the rim, which is exactly what the circulation counts.
Again the picture steps aside and only two numbers remain. The left one is ∫∫ ∇×F dA, the area integral of every bit of curl inside the circle. The right one is ∮ F·dr, the circulation of the field once around the rim. Move the radius slider and the two values grow together and stay nearly equal, and the match mark in the middle confirms it. Where Gauss's theorem traded the divergence of a region for the flux through its boundary, Stokes's theorem trades the curl of a surface for the circulation around its boundary. Add up all the turning inside and you get the turning of the boundary.
Last, the two theorems share one circle. There is one pair of buttons. Press Gauss and outward normal arrows appear on the rim, and the label becomes flux out equals the sum of divergence inside. Press Stokes and the same rim switches to tangent arrows, and the label becomes circulation equals the sum of curl inside. One control swaps ∇· for ∇×, and swaps the normal for the tangent, all at once. The two shapes are twins. Both say that a derivative added up inside equals an integral measured on the boundary. Only the choice of derivative, and the way the arrows stand on the boundary, changes. That one sentence carries vector calculus.