Theorems
Likewise, there are several important theorems related to these operators which generalize the fundamental theorem of calculus to higher dimensions:
| Theorem | Statement | Description |
|---|---|---|
| Gradient theorem | The line integral through a gradient (vector) field equals the difference in its scalar field at the endpoints of the curve L. | |
| Green's theorem | The integral of the scalar curl of a vector field over some region in the plane equals the line integral of the vector field over the closed curve bounding the region. | |
| Stokes' theorem | The integral of the curl of a vector field over a surface in equals the line integral of the vector field over the closed curve bounding the surface. | |
| Divergence theorem | The integral of the divergence of a vector field over some solid equals the integral of the flux through the closed surface bounding the solid. |
Read more about this topic: Vector Calculus