Question

2. Let U C R2 be simply connected and let to E U. Let g: U(oR2 be irrotational and of class C1. Assume that there exists r >0 such that B(zo, r) C U and g=0. Let γ be a closed sinile polygonal arc with range in U \ {zo), let「be its range, and let V be the bounded connected component of R2 \ Г. (a) Assume that V C U \ [xo) and prove that g=0. (b) Assume that o E V and prove that g=0. e) Prove that g is conservative in U\(o) (d) Can you prove (a) without using the part we did on complex func- tions?

Answer 1

Solution: 2 .Let
U C
be simply connected and let
TO EU
. Let $g: U $\textbackslash $ \{x_{0}\}\to R^2$

be irrotational and of class $C^1$. Assume that there exists r>0 such that $\overline{B(x_{0},r)}\subset U$

and $\int _{\delta B(x_{0},r)}g=0$

Let $\gamma$ be a closed simple polygonal arc with range in
Uro)
, let $\Gamma$ be its range and let V be the bounded connected component of $R^2 $ \textbackslash $ \Gamma$ .

(a) Assume that $V\subseteq U $\textbackslash $ \{x_{0}\}$ .

Wwe know that $\oint _{C}\bf{f}\cdot d\bf{r}=0$ for every simple closed curve C in a simply connected

solid region S.

Since $\gamma$ is a closed simple polygonal (closed) arc with range in
Uro)
,

therefore $\int _{\gamma}g=0$ .(Proved)

(c) A conservative vector field is one which has a potential.

We know that "If a vector field $\bf{f}$ has a potential in a region R, then $\oint _{C}\bf{f}\cdot d\bf{r}=0$ for any closed curve C in R .(That is
Vf.dr = 0
for any real-valued function F(x,y))".

Since $\gamma$ is a closed simple polygonal (closed) arc with range in
Uro)
,

and $\int _{\gamma}g=0$ (by a)

Therefore g is conservative. (Proved)