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\begin{document}
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\section*{ 1. Integration of n-forms on oriented M.}

{\bf Recall} : Change of variable formula.\\



$\int_{\vphi(D)}fdx_1\cdot\cdot\cdot dx_n=\int_D
f\circ\vphi|det\vphi_*|du_1\cdot\cdot\cdot du_n$.\\

If $\vphi$ is orientation-preserving,
$\int_{\vphi(D)}fdx_1\cdot\cdot\cdot dx_n=\int_Df\circ\vphi
\,\,(det\,\vphi_*)du_1 \cdot\cdot\cdot du_n$.\\

$\rb^n$������ n-form $\ome=fdx_1\we\cdot\cdot\cdot\we dx_n$��
�����غ���. $\rb^n$���� $n$-form�� ���� ������ $\int\ome=\int
fdx_1\we\cdot\cdot\cdot \we dx_n:=\int fdx_1\cdot\cdot\cdot
dx_n$ ���� �����Ѵ�. �׷���\\

$\vphi^*\ome=\vphi^*(f)\vphi^*(dx_1)\we\cdot\cdot\cdot\we\vphi^*(dx_n)$

$\hs{2em}=(f\circ\vphi)d(\vphi^*x_1)\we\cdot\cdot\cdot\we
d(\vphi^*x_n)$

$\hs{2em}=(f\circ\vphi)d(x_1\circ\vphi)\we\cdot\cdot\cdot
d(x_n\circ\vphi)$

$\hs{2em}=(f\circ\vphi)det(\frac{\pa x}{\pa
u})du_1\we\cdot\cdot\cdot\we du_n\hs{2em}$ (note that
$d(x_i\circ\vphi)=\sum_j\frac{\pa x_i}{\pa u_j} du_j )$

$\hs{2em}= f\circ\vphi\,\,(det\vphi_*)du_1\we\cdot\cdot\cdot\we
du_n$.\\

���� �� ������ȯ ������ ������ ���� ������ ǥ���� �� �ִ�.\\


$(*)\hs{4em}\dis{\int_{\vphi(D)}w=\int_D\vphi^*w}$.\\

If $\vphi$ is orientation-reversing, then
$\hs{1em}\dis{\int_{\vphi(D)}w=-\int_D\vphi^*w}$.\\

���� manifold �������� n-form�� ������ ��������. ���� $\ome$��
$supp\,\,\ome\subset(U,x)$,

a positive coordinate chart(i.e., $x$�� orientation preserving )��
�����ϴ� $n$-form�̶�� ����. (�� single chart�� ���Եȴ�.) ��
$\ome$�� ���� ������ ������ ���� �����Ѵ�.

$\dis{\int_M\ome:=\int_{\widetilde{U}}\vphi^*\ome}$,
$\hs{2em}\vphi=x\inv,\widetilde{U}=x(U)$.

$\dis{(\int_M\ome:=-\int_{\widetilde{U}}\vphi^*\ome},\hs{1.8em}$
if
$\vphi$ is orientation-reversing.)\\

�� ���ǰ� �� ���ǵ��� ������. coordinate chart�� ���ÿ� ��������
���̱� ���� $(U,y)$�� �ٸ� positive coordinate chart ��� ����.\\




$\widetilde{\widetilde{U}}=y(U),\,\,\psi=y\inv$��� �θ�,
transition map $x\circ\psi$�� $\rb^n$���� �Լ��� �ǰ� ���⿡
������ȯ ���� $(*)$ �� �� �� �ִ�. �׷���

$\dis{\int_{\widetilde{U}}\vphi^*\ome=\int_{\widetilde{\widetilde{U}}}(x\circ\psi)^*(\vphi^*\ome)}=
\int_{\widetilde{\widetilde{U}}}\psi^*\ome$,
$\hs{1em}(x\circ\vphi=x\circ x\inv=id$�̹Ƿ�)



$\therefore\,\,\,\dis{\int_{\widetilde{U}}\vphi^*\ome=\int_{\widetilde{\widetilde{U}}}\psi^*\ome}$.\\

���� �Ϲ����� $\ome\,\,(n$-form with compact support on orientable
$M$) �� ���ؼ� ������ ��������. ���� finite cover
$\mathcal{U}=\{U_i\}$ of positive coordinate charts��
$\mathcal{U}=\bigcup U_i\supset $supp$ \,\,\ome$ �� �����ϰԲ�
��´�. $\{\rho_i\}$ �� partition of unity(subordinate
to $\mathcal{U}=\{U_i\}$ )�� �ΰ� ������ ���� ������ �����Ѵ�.\\

$\hs{2em}\dis{\int_M\ome:=\sum_i\int_M\rho_i\ome}$.

���� ���ǰ� cover $\mathcal{U}$�� partition of unity �� ���ÿ�
�����ϰ� �� ���ǵ��� ������. $\mathcal{V}=\{V_j\},\{\tau_j\}$��
�Ǵٸ� cover,partition of unity��� ����. �׷���

$\dis{\sum_j\int_M\tau_j\ome=\sum_j\sum_i\int_M\rho_i(\tau_j\ome)\hs{1em}}$(������ ���ǿ� ����)\\

�� $\rho_i\ome$���� single coordinate $ U_i$�ȿ� �����Ƿ� additive
�̴�. ���� �� ����

$\dis{\sum_j\int_M\tau_j\ome=\sum_i\int_M(\sum_j\tau_j)\rho_i\ome=\sum_i\int_M\rho_i
M.}$\\

{\bf Exercise.}

(1) $\mathcal{E}_c^k(M)$=the space of differentiable $k$-forms on
$M$ with compact support.

Then $\hs{1em}\dis{\int_M:\mathcal{E}_c^k(M)\rightarrow\rb}\hs{1em} $ is linear.\\

(2) $\vphi:M\rightarrow N$, a diffeomorphism. Then

$\dis{\int_M\vphi^*\ome=\pm\int_N\ome}\hs{1em}$($+$: if $\vphi$ is
orientation-preserving,
$-:$ if $\vphi$ is orientation-reversing.)\\

(3) $\vphi:\widetilde{M}\rightarrow M$, an n-fold covering with
induced orientation. Then

$\hs{3em}\dis{\int_{\widetilde{M}}\vphi^*\ome=n\,\int_M\ome}$.




\end{document}
