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

## Comments 0

Log in to post a comment