k¥6 Orientation :

. Stokes'

theorem C 16.8 ) walk along C,

head in direction of It

Recall : If E =D , Q , R ) ,then

.no Surface to your left

.

dirt E) =D . E' = Pat Qyt Rz

cancer .one :p:Iaf

P Q R -

= ( Ry - Qz,Pz - Ra

,Qa - Py ) Right - hand rule :

-

Nxt

stokesltheorem

- y .

IfcF. de =ffscurlCEJ.DE/ n

-

NL

E'

vector field NIF points intos surface with the surface .

C boundary curve}compatible -

orientations

÷:÷i¥

Example : E

=Hz

, yz , xy > RelationshipwithGreen'stheorem_

T YS

zaS flat ,

c,

YEZZG /.

,g=zin xy - plane .

it.

--

-

.

l I yiii.-- -

-it÷ 1¥\

-- -

-

-

-

Ky 't^

>

-- -

- -

-

-

-

ffsoxE.ds.fr?dp=ofcEidF=ffsEELkdAQa-Py#

On C : x±yft2-2=4

FdvelocityI

⇒ z=VI dz=O

×= cost ) dx= - sin # Idf µ✓ ONE .D= 2W

y-

- sink ) dy -

- cosltldt

,-

-

-

-- -

- -

-

-

f. E . DF -

-

fccxziyzixys.sdxidy.dz

) ¥,=f ! cost

, B Sint ,sint . cost > . E

-

C- sint , cost , o >

dtffsoxf.Nds-fcf.fdshdfhdf-f.TV#tsinttr3sint-cos

0xFiN=2w+ o ) It =2( normal component of

=O¢ angular velocity)E. F = tangential component of

velocity .

surfaceindepencei-n.tk :

SAE¥7s

ffs,

0xFiNdS-fcF.dFIL-ffsz@xfT.Nds

-

II

Another way to get that IFIZ :

Ii - Ia =ff WE'

. DdsSi - Sz

= fffeo.CO/lE)dV-- OT Forfar any

E'

Divergence theorem

E- solid region enclosed by She Sz.