\magnification 1800
\parskip 10pt
\parindent 0pt
\hoffset -0.3truein
\voffset -0.5truein
%%\hsize 6.7truein

\voffset -0.8truein
\vsize 9.8truein

\def\emph{}

\def\S{\Sigma_{i=1}^n}
\def\Sm{\Sigma_{i=1}^m}

\def\s{{\sigma}}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\h{\hskip 10pt}
\def\u{\vskip -8pt}
\def\bh{\hfil \break}




\def\N{{\bf N}}
\def\S{\Sigma_{i=1}^n}

\def\0{{\bf 0}}
\def\Z{{\bf Z}}
\def\R{{\bf R}}
\def\C{{\bf C}}
\def\x{{\bf x}}
\def\a{{\bf a}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\n{{\bf n}}
\def\k{{\bf k}}
\def\p{\psi}
\def\f{\phi}
\def\s{\sigma}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\u{ \vskip -5pt }
\def\h{ \hskip 20pt}
\def\v{ \vskip 5pt }


{\bf 6.2 Green's Theorem}

Suppose $F: \R^2 \rightarrow \R^2$, $F(x, y) = (M(x, y), N(x, y))$.
\hfil \break
$S \subset \R^2$ is a nice compact
surface.

$\int_{\partial S} F \cdot ds = \int_{\partial S} (M, N) \cdot (dx, dy) =$

$\int_{\partial S} Mdx + Ndy  = \int\int_S [{\partial N \over \partial x} - {\partial M \over \partial y}]dA$
$ =\int\int_S [\nabla \times F] \cdot \k dA$


\vfill
\hrule

{\bf 7.3 Stoke's Theorem}

Suppose $F: \R^3 \rightarrow \R^3$, $F(x, y, z) = (M(x, y, z), N(x, y, z), P(x, y, z))$. $S \subset \R^3$ is a bounded, piecewise smooth, oriented
surface.

$\int_{\partial S} F \cdot ds  =\int\int_S [\nabla \times F] \cdot dS$

\v

\rightline{$=\int\int_X [(\nabla \times F) (X(s, t))] \cdot N(s, t) ds dt$}

\vfill

\eject

Ex:  Suppose $\partial S = \emptyset$

$\int\int_S [\nabla \times F] \cdot dS  = \int_{\partial S} F \cdot ds  = $




\vfill
\hrule


Ex:  Suppose $\partial S = \{(x, y, 0) ~|~ x^2 + y^2 = 1\} = \partial {\tilde S}$


$\int\int_S [\nabla \times F] \cdot dS  = \int_{\partial S} F \cdot ds  = $
$\int\int_{\tilde S} [\nabla \times F] \cdot d{\tilde S}$



Suppose ${\tilde S} = \{(x, y, 0) ~|~ x^2 + y^2 \leq 1\}$ 

A parametrization for ${\tilde S}$ 

$X(s, t) = (s (cos t), s (sin t), 0)$

$T_s(s, t) = (cos t, sin t, 0)$

$T_t(s, t) = (-s (sin t), s (cos t), 0)$

$N(s, t) = (0, 0, s)$

Suppose $F(x, y, z) = (y^2, 3, z^2)$

\vfill
\eject
~

\eject

Another parametrization for ${\tilde S}$ 

$Y(s, t) = (s , t, 0)$

$T_s(s, t) = (1, 0, 0)$

$T_t(s, t) = (0, 1, 0)$

$N(s, t) = (0, 0, 1)$

Suppose $F(x, y, z) = (y^2, 3, z^2)$

$\nabla \times F = (0, 0, -2y)$

$\int\int_{\tilde S} [\nabla \times F] \cdot d{\tilde S}$
$ = \int\int_{\tilde S} (0, 0, -2y) \cdot d{\tilde S}$

{$=\int\int_Y [(\nabla \times F) (X(s, t))] \cdot N(s, t) ds dt$}

{$=\int_{-1}^1\int_{-\sqrt{1 - t^2}}^{\sqrt{1 - t^2}}(0, 0, -2t) \cdot (0, 0, 1) ds dt$}

{$=-\int_{-1}^1\int_{-\sqrt{1 - t^2}}^{\sqrt{1 - t^2}} ~2t ds dt$}

{$=-\int_{-1}^1 2ts|_{-\sqrt{1 - t^2}}^{\sqrt{1 - t^2}} ~ dt$}


{$=-\int_{-1}^1 4t \sqrt{1 - t^2} ~ dt$}

Let $u = 1 - t^2$, $du = -2tdt$, $t = \pm 1$ implies $u = 0$

{$=-\int_{0}^0 -2 \sqrt{u} ~ du = 0$ }

\eject

{\bf 6.2 Divergence Theorem in the Plane}

Suppose $F: \R^2 \rightarrow \R^2$, $F(x, y) = (M(x, y), N(x, y))$.
\hfil \break
$S \subset \R^2$ is a nice compact
surface. \hfil \break
$\n \subset \R^2$, outward unit normal to $\partial S$.




$ \int_{\partial S} F \cdot \n  ds  = $
$\int\int_{S} [\nabla \cdot F] dA$




\vfill
\hrule

{\bf 7.3 Gauss's Theorem}

Suppose $F: \R^3 \rightarrow \R^3$, $F(x, y, z) = (M(x, y, z), N(x, y, z), P(x, y, z))$. 
$D \subset \R^3$ is a bounded, solid nice 3-dimensional region.

$\int \int_{\partial D} F \cdot dS  =\int \int\int_D [\nabla \cdot F] dV$


\vfill
\vfill

\eject



Calculate $\int \int_S F \cdot dS$ where 

$F(x, y, z) = (xy^2, y^3, 4x^2z)$, ~~ $S = S_1 \cup S_2 \cup S_3$ and

$S_1 = \{ (x, y, 5) ~|~ x^2 + y^2 \leq 4 \}$ 

$S_2 = \{ (x, y, z) ~|~ x^2 + y^2 = 4, 0 \leq z \leq 5 \}$ 



$S_3 = \{ (x, y, 0) ~|~ x^2 + y^2 \leq 4 \}$ 


\vfill

\eject

Let $S_a$ = sphere of radius $a$ centered at the point $P = (P_1, P_2, P_3)$


Prop 3.4.  Divergence of $F$ at $P$= 

$(\nabla \cdot F)(P_1, P_2, P_3)
=$
$ lim {3 \over 4\pi a^3} \int\int_{S_a} F \cdot dS$

$ = lim_{a \rightarrow 0^+} {\int\int_{S_a} F \cdot dS \over {4\pi a^3 \over 3}} $

$ = lim_{a \rightarrow 0^+}  {\hbox{flux} \over {\hbox{volume of ball of radius a}}} $
 = flux density


\vfill
\eject

Let $C_a$ = the circle of radius $a$ centered at the point $P = (P_1, P_2, P_3)$ lying in the plane perpendicular to the unit vector $\n$ and containing the point $P$. 



Prop 3.5.  The component of the curl of $F$ at $P$ in the direction of $\n = $

%%||$proj$_\n (\nabla \times F)(P)|| =$

$\n \cdot (\nabla \times F)(P) = 
lim_{a \rightarrow 0^+} {1 \over \pi a^2} \int_{C_a} F \cdot ds$

$= lim_{a \rightarrow 0^+} {1 \over \pi a^2} \int_X (F \cdot T) ds$

$= {\hbox{circulation of $F$ along $C_a$} \over \hbox{area of surface bounded by $C_a$}}$










\end