Theorem:
Let
Stokes’ Theorem (general version)
S be an oriented k–dimensional surface in Rn and let Φ be
a closed (k − 1)–dimensional surface contained in
S . Let S be a
surface enclosed by Φ (Φ = ∂S).
If ω is a (k − 1)–form, then
Z
dω =
S
Z
Φ=∂S
ω
provided the orientations on S and ∂S are compatible.
STOKES’ THEOREM
n≤3 ,
n=1
k=1
k≤n
FTC
Z
∇f ·ds = f (end) − f (start)
n = 2, 3 k = 1
GF T C
n=2
k=2
Green
n=3
k=3
Divergence(Gauss)
n=3
k=2
γ
For n = 3, k = 2, let ω = F1 dx + F2 dy + F3 dz then
dω = dF1 ∧ dx + dF2 ∧ dy + dF3 ∧ dz
∂F1
∂F1
∂F1
∂F2
∂F2
=
dx +
dy +
dz ∧ dx +
dx +
dy +
∂x
∂y
∂z
∂x
∂y
∂F3
∂F3
∂F2
∂F3
dz ∧ dy +
dx +
dy +
dz ∧ dz
∂z
∂x
∂y
∂z
∂F1
∂F2
∂F2
∂F1
dy ∧ dx +
dz ∧ dx +
dx ∧ dy +
dz ∧ dy +
=
∂y
∂z
∂x
∂z
∂F3
∂F3
dx ∧ dz +
dy ∧ dz
∂x ∂y ∂F1
∂F2
∂F2
∂F3
=
−
dx ∧ dy +
−
dy ∧ dz +
∂x
∂y
∂y
∂z
∂F1 ∂F3
dz ∧ dx
−
∂z
∂x
If F = (F1 , F2, F3), Stokes’ Theorem (n = 3, k = 2) gives
Z
Z
Z
F ·ds =
ω = dω
∂S
∂S
S
Z ∂F3
∂F2
∂F1
∂F3
−
dy ∧ dz +
−
dz ∧ dx
=
∂y
∂z
∂z
∂x
S
∂F2 ∂F1
−
dx ∧ dy
+
∂x
∂y
Z ∂F3 ∂F2 ∂F1 ∂F3 ∂F2 ∂F1
=
−
,
−
,
−
·dS.
∂y
∂z
∂z
∂x
∂x
∂y
S
Definition:
If F = (F1 , F2, F3, ), the vector field
∂F3 ∂F2 ∂F1 ∂F3 ∂F2 ∂F1
−
,
−
,
−
∂y
∂z ∂z
∂x ∂x
∂y
is calld the curl of F and is denoted curl F or ∇ × F .
Theorem:
If f : R3 → R is of class C 2, then
∇ × (∇f ) = 0
curl ( grad f ) = 0
Theorem:
If F is a C 2 vector field on R3 then
∇·(∇ × F ) = 0
div ( curl F ) = 0
Remark:
If div G = 0 and G is defined throughout R3, then there
is a F such that curl F = G.
If there is a F such that curl F = G, then F is said to be a vector
potential for G.
If F is a vector potential for G, then so is F +∇h, where h : R3 → R
is of class C 2 (the gauge freedom).
Theorem: If S is a closed bounded (2–dim) surface in R3 and F is
a C 2 vector field defined throughout the interior, R, of S then
R
S=∂R ( curl F
)·dS = 0
Assume that G has a vector potential F . If oriented surfaces S1 and
S2 have the same compatibily oriented boundary C, then
Z
Z
Z
G·dS =
G·dS
=
F ·ds .
S1
S2
C
If G has a vector potential, it is surface independent (just as a vector
field with scalar potential is path independent).
Definition:
If curl F = 0 at a point, the flow is said to be
irrotational at the point (i.e., no whirlpool).
Theorem:
Let
Stokes’ Theorem (special case)
S be an oriented (2–dimensional) surface in R3 and let γ be an
oriented simple closed curve on
S . Let S be a surface enclosed by
γ (γ = ∂S). If F is a C 1 vector field on
Z
F ·ds =
γ=∂S
Z
S , then
( curl F )·dS
S
provided the orientations are compatible.