A Proof of Stoke's Theorem   

The proof of Stoke's theorem follows directly from Green's theorem. Before doing so, however, let's notice that if F = á M,N,P ñ is a vector field and if we let F₁ = á M,0,0 ñ , F₂ = á0,N,0 ñ , and F₃ = á 0,0,P ñ , then F = F₁+F₂+F₃ and similarly

curl( F) = curl( F1) +curl( F2) +curl( F3)

As a result, Stoke's theorem is proved if we can show it for each of F₁, F₂, and F₃. Below we prove Stoke's theorem for F₁ = á M,0,0 ñ . Proofs for F₂ and F₃ are left to the exercises.

Suppose that r( u,v) = á x( u,v), y( u,v), z( u,v) ñ which maps a region S in the uv-plane to a surface S in R³, and suppose also that the boundary of S is mapped to the boundary of S. If ¶S is parameterized by r( t) for t in [ a,b] , then the work integral becomes

¶S  F1 · dr = ó õ b a  F1 ·   dr dt dt = ó õ b a  F1 ·  æ è  ¶r ¶u  du dt +  ¶r ¶v  dv dt ö ø dt = ó õ b a  F1 ·  ru    du dt +F · rv  dv dt   dt = ¶S  F1 · ru  du+F1 · rv  dv

That is, we can pull the boundary curve back into the uv-plane. However, Green's theorem in the uv-plane implies that

¶S  F1 · ru  du+F1 · rv  dv =   ó õ ó õ S    ¶ ¶u ( F1 · rv) -  ¶ ¶v ( F1 · ru)   dudv

If we let F_1,u denote the partial of F₁ with respect to u - i.e., F_1,u = ¶_uF₁  - then

¶S  F1 · dr = ó õ ó õ S  ¶ ¶u ( F1· rv)  -   ¶ ¶v ( F1· ru)  dudv = ó õ ó õ S F1,u · rv+F1 · ruv-F1 · ruv-F1,v · rv  dudv = ó õ ó õ S F1,u · rv-F1,v · ru  dudv

Substituting F₁ = á M,0,0 ñ , r_u = á x_u, y_u, z_u ñ , and r_v = á x_v, y_v, z_v ñ results in

¶S  F1 · dr = ó õ ó õ S áMu,0,0 ñ · áxv,yv,zv ñ - á Mv,0,0 ñ · á xu,yu,zu ñ   dudv = ó õ ó õ S ( Muxv-Mvxu) dudv

Our goal now is to show that the integrand of the double integral is curl( F₁) · dS. To do so, we first use the chain rule in the form M_u = M_xx_u+M_yy_u+M_zz_u and M_v = M_xx_v+M_yy_v+M_zz_v to obtain

¶S  F1 · dr = ó õ ó õ S  [( Mxxu+Myyu+Mzzu) xv-(Mxxv+Myyv+Mzzv) xu] dudv = ó õ ó õ S  [ ( Myyu+Mzzu) xv-(Myyv+Mzzv) xu] dudv = ó õ ó õ S  [Myyuxv+Mzzuxv-Myyvxu-Mzzvxu] dudv

Rearranging terms and factoring out M_z and M_y yields

¶S  F1 · dr = ó õ ó õ S [Mz( zuxv-zvxu) -My(xuyv-xvyu) ] dudv = ó õ ó õ S Mz ê ê ê zu xu zv xv ê ê ê  - My ê ê ê xu yu xv yv ê ê ê  dudv

Since curl( F₁) = á0,M_z,-M_y ñ , we rewrite the integrand as

¶S  F1 · dr = ó õ ó õ S   á0,Mz,-My ñ ·  ê ê ê yu zu yv zv ê ê ê , ê ê ê zu xu zv xv ê ê ê , ê ê ê xu yu xv yv ê ê ê  dudv = ó õ ó õ S curl( F1) ·( ru×rv) dudv = ó õ ó õ S curl( F1) · dS

The proof for F₂ and F₃ is similar, so that

ó õ ó õ S  curl( F) · dS = ¶S  F1 · dr +  ¶S  F2 · dr +  ¶S  F3·dr = ¶S  ( F1+F2+F3) · dr = ¶S  F · dr

thus proving Stoke's theorem.