Problem

二、多選題

13. A region $A$ is bounded by the hyperbolas（雙曲線）$xy=1$ and $xy=2$, and the curves $xy^2=3$ and $xy^2=4$. The area of $A$ is denoted by $area_o$. Use the change of variables $u=xy$ and $v=xy^2$, the area of the image of the region $A$ in the coordinate space determined by $(u,v)$ under the transformation is called $area_t$. 請問下列哪些選項是正確的？

(A) $area_o=\ln 4-\ln 3$

(B) $area_t=\int_1^2\int_3^4 \mathrm{d}u\mathrm{d}v$

(C) $area_o=\int_1^2\int_3^4 \dfrac{1}{v}\,\mathrm{d}u\mathrm{d}v$

(D) The Jacobian determinant $\dfrac{\partial(x,y)}{\partial(u,v)}$ is $v$.

(E) $\dfrac{\partial(u,v)}{\partial(x,y)}=xy$。