Vector Calculus - Theorems

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.