변수변환과 야코비안

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.

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

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

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

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

$x=u\,\cos ({e}^{v}),y=u\,\sin ({e}^{v})$

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

$x=v\,\sin ({u}^{2}),y=v\,\cos ({u}^{2})$

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

$x=u\,\cosh \,v,y=u\,\sinh \,v,z=w$

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

$x=v\,\cosh \left(\frac{1}{u}\right),y=v\,\sinh \left(\frac{1}{u}\right),z=u+{w}^{2}$

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

$x=u+v,y=v+w,z=u$

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

$x=u-v,y=u+v,z=u+v+w$

The triangular region $R$ with the vertices $(0,0),(1,1),\text{and}\,(1,2)$ is shown in the following figure.

a. Find a transformation $T:S\to R,$ $T(u,v)=(x,y)=(au+bv,cu+dv),$ where $a,b,c,$ and $d$ are real numbers with $ad-bc\ne 0$ such that ${T}^{-1}(0,0)=(0,0),{T}^{-1}(1,1)=(1,0),$ and ${T}^{-1}(1,2)=(0,1).$ b. Use the transformation $T$ to find the area $A(R)$ of the region $R.$

The triangular region $R$ with the vertices $(0,0),(2,0),\text{and}\,(1,3)$ is shown in the following figure.

a. Find a transformation $T:S\to R,$ $T(u,v)=(x,y)=(au+bv,cu+dv),$ where $a,b,c$ and $d$ are real numbers with $ad-bc\ne 0$ such that ${T}^{-1}(0,0)=(0,0),$ ${T}^{-1}(2,0)=(1,0),$ and ${T}^{-1}(1,3)=(0,1).$ b. Use the transformation $T$ to find the area $A(R)$ of the region $R.$

In the following exercises, use the transformation $u=y-x,v=y,$ to evaluate the integrals on the parallelogram $R$ of vertices $(0,0),(1,0),(2,1),\text{and}\,(1,1)$ shown in the following figure.

$\iint_{R}(y-x)dA$

In the following exercises, use the transformation $u=y-x,v=y,$ to evaluate the integrals on the parallelogram $R$ of vertices $(0,0),(1,0),(2,1),\text{and}\,(1,1)$ shown in the following figure.

$\iint_{R}({y}^{2}-xy)dA$

In the following exercises, use the transformation $y-x=u,x+y=v$ to evaluate the integrals on the square $R$ determined by the lines $y=x,y=-x+2,y=x+2,$ and $y=-x$ shown in the following figure.

$\iint_{R}{e}^{x+y}dA$

In the following exercises, use the transformation $y-x=u,x+y=v$ to evaluate the integrals on the square $R$ determined by the lines $y=x,y=-x+2,y=x+2,$ and $y=-x$ shown in the following figure.

$\iint_{R}\sin (x-y)dA$

In the following exercises, use the transformation $x=u,5y=v$ to evaluate the integrals on the region $R$ bounded by the ellipse ${x}^{2}+25{y}^{2}=1$ shown in the following figure.

$\iint_{R}\sqrt{{x}^{2}+25{y}^{2}}\,dA$

In the following exercises, use the transformation $x=u,5y=v$ to evaluate the integrals on the region $R$ bounded by the ellipse ${x}^{2}+25{y}^{2}=1$ shown in the following figure.

${\iint_{R}({x}^{2}+25{y}^{2})}^{2}dA$