Fundamental Theorems
Catalin Zara
UMass Boston
May 12,2010
Catalin Zara (UMB)
Fundamental Theorems
May 12,2010 1/6
A Unifying Theme
Accumulation of a quantity Accumulation of a
over the boundary of = derived quantity
a closed domain over the entire domain
the domain is oriented;
the orientation of the domain induces an orientation of the boundary.
Catalin Zara (UMB)
Fundamental Theorems
May 12,2010 2/6
Domains of Dimension One
Fundamental Theorem of Line Integrals
:
C:smooth curve,joining points A and B,oriented from A to B;
∂C = {A,B}:boundary of C;
Orientation of ∂C:A,weight 1;B,weight +1;
f a diﬀerentiable function deﬁned on (an open neighborhood of) C.
f(B) −f(A) =
C
df ⇐⇒
∂C
f =
C
df.
r:[a,b] →R
n
,smooth parametrization of C with r(a) = A and r(b) = B:
f(r(b)) −f(r(a)) =
C
∇f dr.
Net Change Theorem
:If f:[a,b] →R is a diﬀerentiable function,then
f(b) −f(a) =
b
a
f
′
(x) dx.
Catalin Zara (UMB)
Fundamental Theorems
May 12,2010 3/6
Domains of Dimension Two
Stokes’ Theorem
:
S:surface,oriented by unit normal ﬁeld n;D:domain on S;
C = ∂D:boundary of D,oriented by the outward unit normal N;
X:is a smooth ﬁeld on D.
∂D
X dr =
D
curlX dS
If X= P(x,y,z) i +Q(x,y,z) j +R(x,y,z) k,then
curlX= h∂
y
R−∂
z
Q,∂
z
P −∂
x
R,∂
x
Q−∂
y
Pi =
i j k
∂
x
∂
y
∂
z
P Q R
= ∇×X.
Green’s Theorem
:Particular case when S is a plane,oriented by k.
∂D
P(x,y)dx +Q(x,y)dy =
D
(Q
x
−P
y
) dxdy
∂D
X dr =
D
curl
k
XdA,
∂D
X nds =
D
div XdA
Catalin Zara (UMB)
Fundamental Theorems
May 12,2010 4/6
Domains of Dimension Three
Divergence Theorem
:
D,domain in R
3
;
∂D:boundary of D,oriented by the outward unit normal n;
X:smooth vector ﬁeld on D.
∂D
X dS =
D
divXdV,
where X dS = X ndS.
If X= P(x,y,z) i +Q(x,y,z) j +R(x,y,z) k,then
div X= P
x
+Q
y
+R
z
= ∇ X.
X= gradf =⇒curl X= curl(gradf) = 0 =⇒condition for scalar potential
X= curl G=⇒div X= div(curl G) = 0 =⇒condition for vector potential
Catalin Zara (UMB)
Fundamental Theorems
May 12,2010 5/6
Higher Dimensional Domains
Accumulation of a quantity Accumulation of a
over the boundary of = derived quantity
a closed domain over the entire domain
the domain is oriented;
the orientation of the domain induces an orientation of the boundary.
General Stokes Theorem
:
∂M
ω =
M
dω
It would take too long to explain here what all that means,
Will be happy to do so in a future course,Analysis on Manifolds.
Catalin Zara (UMB)
Fundamental Theorems
May 12,2010 6/6
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