다중적분 전체 문제

In the following exercises, the function $T:S\to R,T(u,v)=(x,y)$ on the region $S=\{(u,v)|0\le u\le 1,0\le v\le 1\}$ bounded by the unit square is given, where $R\subset {\mathbb{R}}^{2}$ is the image of $S$ under $T.$

a. Justify that the function $T$ is a ${C}^{1}$ transformation. b. Find the images of the vertices of the unit square $S$ through the function $T.$ c. Determine the image $R$ of the unit square $S$ and graph it.

$x={u}^{3},y={v}^{3}$

In the following exercises, determine whether the transformations $T:S\to R$ are one-to-one or not.

$x={u}^{2},y={v}^{2},\text{where}\,S$ is the rectangle of vertices $(-1,0),(1,0),(1,1),\text{and}\,(-1,1).$

In the following exercises, determine whether the transformations $T:S\to R$ are one-to-one or not.

$x={u}^{4},y={u}^{2}+v,\text{where}\,S$ is the triangle of vertices $(-2,0),(2,0),\text{and}\,(0,2).$

In the following exercises, determine whether the transformations $T:S\to R$ are one-to-one or not.

$x=2u,y=3v,\text{where}\,S$ is the square of vertices $(-1,1),(-1,-1),(1,-1),\text{and}\,(1,1).$

In the following exercises, determine whether the transformations $T:S\to R$ are one-to-one or not.

$T(u,v)=(2u-v,u),$ where $S$ is the triangle of vertices $(-1,1),(-1,-1),\text{and}\,(1,-1).$

In the following exercises, determine whether the transformations $T:S\to R$ are one-to-one or not.

$x=u+v+w,y=u+v,z=w,$ where $S=R={\mathbb{R}}^{3}.$

In the following exercises, determine whether the transformations $T:S\to R$ are one-to-one or not.

$x={u}^{2}+v+w,y={u}^{2}+v,z=w,$ where $S=R={\mathbb{R}}^{3}.$

In the following exercises, the transformations $T:S\to R$ are one-to-one. Find their related inverse transformations ${T}^{-1}:R\to S.$

$x=4u,y=5v,$ where $S=R={\mathbb{R}}^{2}.$

In the following exercises, the transformations $T:S\to R$ are one-to-one. Find their related inverse transformations ${T}^{-1}:R\to S.$

$x=u+2v,y=-u+v,$ where $S=R={\mathbb{R}}^{2}.$

In the following exercises, the transformations $T:S\to R$ are one-to-one. Find their related inverse transformations ${T}^{-1}:R\to S.$

$x={e}^{2u+v},y={e}^{u-v},$ where $S={\mathbb{R}}^{2}$ and $R=\{(x,y)|x>0,y>0\}$

In the following exercises, the transformations $T:S\to R$ are one-to-one. Find their related inverse transformations ${T}^{-1}:R\to S.$

$x=\ln \,u,y=\ln (uv),$ where $S=\{(u,v)|u>0,v>0\}$ and $R={\mathbb{R}}^{2}.$

In the following exercises, the transformations $T:S\to R$ are one-to-one. Find their related inverse transformations ${T}^{-1}:R\to S.$

$x=u+v+w,y=3v,z=2w,$ where $S=R={\mathbb{R}}^{3}.$

In the following exercises, the transformations $T:S\to R$ are one-to-one. Find their related inverse transformations ${T}^{-1}:R\to S.$

$x=u+v,y=v+w,z=u+w,$ where $S=R={\mathbb{R}}^{3}.$

In the following exercises, the transformation $T:S\to R,T(u,v)=(x,y)$ and the region $R\subset {\mathbb{R}}^{2}$ are given. Find the region $S\subset {\mathbb{R}}^{2}.$

$x=au,y=bv,R=\{(x,y)|{x}^{2}+{y}^{2}\le {a}^{2}{b}^{2}\},$ where $a,b>0$

In the following exercises, the transformation $T:S\to R,T(u,v)=(x,y)$ and the region $R\subset {\mathbb{R}}^{2}$ are given. Find the region $S\subset {\mathbb{R}}^{2}.$

$x=au,y=bv,R=\{(x,y)|\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}\le 1\},$ where $a,b>0$

In the following exercises, the transformation $T:S\to R,T(u,v)=(x,y)$ and the region $R\subset {\mathbb{R}}^{2}$ are given. Find the region $S\subset {\mathbb{R}}^{2}.$

$x=\frac{u}{a},y=\frac{v}{b},z=\frac{w}{c},$ $R=\{(x,y)|{x}^{2}+{y}^{2}+{z}^{2}\le 1\},$ where $a,b,c>0$

In the following exercises, the transformation $T:S\to R,T(u,v)=(x,y)$ and the region $R\subset {\mathbb{R}}^{2}$ are given. Find the region $S\subset {\mathbb{R}}^{2}.$

$x=au,y=bv,z=cw,R=\{(x,y)|\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}-\frac{{z}^{2}}{{c}^{2}}\le 1,z>0\},$ where $a,b,c>0$

In the following exercises, find the Jacobian $J$ of the transformation.

$x=u+2v,y=-u+v$

In the following exercises, find the Jacobian $J$ of the transformation.

$x=\frac{{u}^{3}}{2},y=\frac{v}{{u}^{2}}$

In the following exercises, find the Jacobian $J$ of the transformation.

$x={e}^{2u-v},y={e}^{u+v}$