MATH2310 – MVC Lecture 11
Stokes’ theorem:
Stokes’ theorem is the higher dimensional version of Green’s Theorem.
Stokes’ theorem relates a surface integral over the surface S to the line integral
along its boundary ∂ S.
Let S be an oriented, piecewise-smooth, surface, bounded by a simple, closed,
piecewise-smooth boundary, ∂ S, with positive orientation. Let F : R3 → R 3 be a vector
field with continuous partial derivatives. Then:
❑ ❑
∫ F dr =∬ curl(F ) dS
∂S S
Suppose ∂ S1=C=∂ S2. Then:
❑ ❑ ❑
∬ curl ( F ) dS=∫ F dr=∬ curl ( F ) dS
S1 C S2
The divergence theorem:
Green’s theorem in its vector (‘div’) form is:
❑ ❑
∫ F ∙ n dS=∬ ¿(F ( x , y ))dA
C D
where D is a simple region and C is its positively oriented boundary curve.
This could be extended to vector fields on R3 so that:
❑ ❑
∬ F dS=∭ ¿(F )dV
S E
where E is a simple solid region, S has outward positive orientation, and F : R3 → R 3 is
a vector field whose components have continuous partial derivatives.
Stokes’ theorem:
Stokes’ theorem is the higher dimensional version of Green’s Theorem.
Stokes’ theorem relates a surface integral over the surface S to the line integral
along its boundary ∂ S.
Let S be an oriented, piecewise-smooth, surface, bounded by a simple, closed,
piecewise-smooth boundary, ∂ S, with positive orientation. Let F : R3 → R 3 be a vector
field with continuous partial derivatives. Then:
❑ ❑
∫ F dr =∬ curl(F ) dS
∂S S
Suppose ∂ S1=C=∂ S2. Then:
❑ ❑ ❑
∬ curl ( F ) dS=∫ F dr=∬ curl ( F ) dS
S1 C S2
The divergence theorem:
Green’s theorem in its vector (‘div’) form is:
❑ ❑
∫ F ∙ n dS=∬ ¿(F ( x , y ))dA
C D
where D is a simple region and C is its positively oriented boundary curve.
This could be extended to vector fields on R3 so that:
❑ ❑
∬ F dS=∭ ¿(F )dV
S E
where E is a simple solid region, S has outward positive orientation, and F : R3 → R 3 is
a vector field whose components have continuous partial derivatives.
