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\begin{document}
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\section*{ 3. Classical versions in $\rb^n$.}

{\bf 1. dim $M$=1 and orientable.}\\

**�׸� 9-1**\\

Stokes ���� $\int_Md\ome=\int_{\pa M}\ome$ ���� $\ome$�� 0-form
�̹Ƿ� �Լ��̴�. �� $f:M\rightarrow\rb$ �� ǥ���ϸ�
$\dis{\int_Mdf=\int_{\pa M}f}$ �� �ȴ�. $\pa M$ �� �� �׸����� ��
�� $P,Q$�� �ǰ� ���⿡ orientation�� �����ؼ� $\int_{\pa
M}f=f(Q)-f(P)$�� �����Ѵ�. ������ $M$��
$\sigma:\rb\rightarrow\rb^3$���� �־����� ��� �̸� Ȯ���� ��
�ִ�.


$\dis{\int_Mdf=\int_{[a,b]}\sigma^*(df)=\int_a^b\sigma^*(df)}$

$\dis{\hs{2.7em}=\int_a^bd(f\circ\sigma)=\int_a^b\frac{d}{dt}
(f\circ\sigma)dt=(f\circ\sigma)(b)-(f\circ\sigma)(a)}$

$\hs{2.7em}=f(Q)-f(P)$\\

�̴� ���������� �⺻�����͵� ����.\\

{\bf 2. dim $M=2,M\subset\rb^2$ with standard orientation.}\\

**�׸� 9-2**\\

���⼭�� $\ome$�� $\rb^2$������ 1-form�̰� �̸� $\ome:=Pdx+Qdy$��
�� �� �ִ�. ���� ��꿡 ���� $\dis{d\ome=(\frac{\pa Q}{\pa
x}-\frac{\pa P}{\pa y})dx\we dy}$ ���� �� �� �ְ� �׷���
$\rb^2$������ Stokes ������ ������ ���� ǥ���Ǵµ� �̴� Green ������ ������ �˼� �ִ�.\\

$\hs{8em}\dis{\int_M(Q_x-P_y)dxdy=\int_{\pa M}Pdx+Qdy}$\\

{\bf 3. dim $M=2,M\subset\rb^3$}.\\

**�׸� 9-3**\\

���� $\rb^3$�� standard orientation����
�ְ� $ M$�� orientation�� normal vector�� �������� ����.\\

$\ome$�� 1-form�� ��� $\ome=Pdx+Qdy+Rdz$ �� �θ� ���� ��꿡 ����

$d\ome=(R_y-Q_z)dy\we dz-(P_z-R_x)dx\we dz+(Q_x-P_y)dx\we dy$ �̰�

���� $\rb^3$������ ������ ���� ǥ���ȴ�.

$\dis{\int_M (R_y-Q_z)dy\we dz-(P_z-R_x)dx\we dz+(Q_x-P_y)dx\we
dy=\int_{\pa M} Pdx+Qdy+Rdz}$.\\

�̰��� �������� å���� ���� ������� �� �ٲٷ��� ������ ���� �ϸ�
�ȴ�.

$"d\sigma"$�� $M$���� Riemannian volume form�̶� ����. �ٽ� ����
�� �� $p\in M$���� $(v_1,v_2)$�� $T_p M$�� positive orthonormal
basis�� �ΰ� $(\eps_1,\eps_2)$�� �̰��� dual basis�� �ξ��� ��
$"d\sigma":=\eps_1\we\eps_2$�� �����Ѵ�. �� �� $\nu$�� ���Ǵ�
positive orthonormal basis�� ���ÿ� �����ϴ�.({\bf Exercise})

�Ϲ������� � 2-form $\eta=ady\we dz-bdx\we dz+cdx\we dy$��
�־����� ������ $f$�� ���� $\eta=f"d\sigma"$�� �� �� �ְ� vector
field $G=(a,b,c)$�� �θ�


$$ f=\eta(v_1,v_2)=\left|\begin{array}{1} a\hs{1em}b\hs{1em}c
\\ v_1\\v_2 \end{array} \right|=G\cdot(v_1\times v_2)=G\cdot N $$





���� ǥ���� �� �ִ�. ���⼭ $N=v_1\times v_2$�� $M$����
orientation�� ���� ������ unit normal vector field�� �ȴ�.

Ư�� �� �Ŀ��� $N=(a_1,a_2,a_3)$�� �θ� $G$ ��� $N$�� ����Ͽ�
$"d\sigma"=(N\cdot N)"d\sigma"=a_1dy\we dz-a_2dx\we dz+a_3dx\we
dy$�� �ȴ�.\\

���� $\ome=Pdx+Qdy+Rdz$, vector field $F=(P,Q,R)$�� �θ�

$d\ome=(Q_x-P_y)dx\we dy-(P_z-R_x)dx\we dz+(R_y-Q_z)dy\we dz$

$\hs{1.2em}:=adx\we dy-bdx\we dz+cdy\we dz$

�� �θ� $\nabla\times F=(a,b,c)$�� �ǹǷ� ������
$d\ome=(\nabla\times F)\cdot N"d\sigma"$ �� �ǰ�

$\dis{\int_Md\ome=\int_M(\nabla\times F)\cdot N"d\sigma"}$ ��
�ȴ�.\\

���� $\pa M$�󿡼� induced orientation �������� unit tangent
vector�� $T$�� �θ� �̰��� dual $"ds"$�� Riemannian orientation
volume form�� �ǰ� $\ome=g"ds"$�� �θ� $g=\ome(T)=F\cdot T$�� �Ǿ�
$\dis{\int_{\pa M}\ome=\int_{\pa M}(F\cdot T)"ds"}$ �� �ȴ�.
���� Stokes ������ ������ ���� ǥ���� ���� �ִ�.\\

$\dis{\int_M\nabla\times F\cdot Nd\sigma=\int_{\pa M}F\cdot
Tds}$.\\\\


{\bf 4. dim $M=3, M\subset\rb^3$}.\\

**�׸� 9-4**\\

$\ome$�� 2-form�� ��� $\ome=ady\we dz-bdx\we dz+cdx\we dy$�� �θ�

$d\ome=(a_x+b_y+c_z)dx\we dy\we dz$ �� �ȴ�. �̸� Stokes ������
�����ϸ�

$\dis{\int_M(a_x+b_y+c_z)dx\we dy\we dz=\int_{\pa M}ady\we
dz-bdx\we dz+cdx\we dy}$.\\

���� ���� vector notation���� ǥ������. �� $F=(a,b,c)$�� �θ�
�տ��� $\ome=F\cdot N"d\sigma"$�� �ǰ� $d\ome= (div F)dx\we dy\we
dz$�� �Ǿ� Stokes ������ ������ ���� ǥ���ȴ�.

$\dis{\int_M (div F)dx dy dz=\int_{\pa M}F\cdot N"d\sigma"}$\\

�̴� �߻�����(Gauss ����)���� �� �� �ִ�.



\end{document}
