APPLICATIONS OF STOKES THEOREM
1. Fundamental Theorem of Algebra
Theorem 1.1. Let p(z ) = z n + an1 z n1 + . + a0 be a monic polynomial of degree
n 1. Then there exists z C so p(z ) = 0.
Proof. We assume, to the contrary, that p(z ) = 0 for all
GREENS, GAUSSS, AND STOKES THEOREM
Here are some notes for Math 4/515 at the University of Oregon in Spring 2010.
In Section 2, we present Greens theorem, Gausss theorem, and Stokes theorem
as they are classically present
9. Homework #9 Math 4/514
We introduce the following translation table in R3 :
(1) A 0-form 0 corresponds to the function f .
(2) A 1-form P dx + Qdy + Rdz corresponds to the vector eld F1 := (P, Q, R).
(3) A 2-form P dy dz + Qdz dx + Rdx dy corresponds t
The Change of Variable Theorem
We wish to prove:
Theorem 1.1. Let O be an open subset of Rn . Let : O Rn be C 1 . Assume
that is 1 1 and that det( (x) = 0 for all x O. Set U = (O); this is an open
subset as well. Let f be integrable in the
1. Positive Function
Let O be an open subset of Rn .
Denition 1.1. Let f be a non-negative function on O.
(1) Support(f ) = Closurecfw_x : f (x) = 0.
(2) f is locally bounded on O means that every point x O has a neighborhood
Ux so that