A Trinity of Theorems

There are three essential theorems that capture the fundamentals of multivariable calculus. (And no, I'm not referring to the fundamental theorem of calculus. That's far to simple!)

1. Green's Theorem
Green's theorem essentially says that you can take any positively oriented closed curve in the plane and rewrite it as a double integral over the area enclosed by the curve. In other words:

2. Divergence Theorem
The divergence theorem says that you can take any non-closed surface, S, in space with a boundary, C, and describe the flux across the surface as a triple integral of the divergence in the volume enclosed by the surface.

3. Stokes' Theorem
And to wrap everything together, Stokes' Theorem says that a loop integral on the boundary, C, of a non-closed surface, S, can be rewritten as a double integral of the curl of a vector field, F, across the surface.

These three theorems can be said to be "fundamental" because they relate the boundaries of surfaces to the interior of those surfaces, just as in basic calculus, the "interior" of a one-dimensional curve in the plane can be related to the endpoints of that curve.