Mapping $\mathit{R}\mathbf{\prime }\mathbf{\to }\mathit{R}$ 

Mapping $\mathit{R}\mathbf{\to }\mathit{R}\mathbf{\prime }$ 

$x\=\frac{v}{\sqrt{-2u\+2\sqrt{u^2\+v^2}}}$

$u\=x^2-y^2$

$y\=\frac{1}{2}\sqrt{-2u\+2\sqrt{u^2\+v^2}}$

$v\=2 x y$

Table 5.6.4(a)   Mappings $R\leftrightarrow R\prime$

$\int {\int }_{R}f\left(x\,y\right) \mathrm{dA}$

= ${\int }_1^4{\int }_1^5\frac{f\left(x\left(u\,v\right)\,y\left(u\,v\right)\right)}{4\sqrt{v^2-u^2}} \mathit{\ⅆ}v\mathit{\ⅆ}u$ 

Initialize

Loading Student:-MultivariateCalculus

$L\=\left[x^2-y^2\=1\, x^2-y^2\=4\,2 x y\=1\,2 x y\=5\right]$$\stackrel{\text{assign}}{\to }$

$U\=x^2-y^2$$\stackrel{\text{assign}}{\to }$

$V\=2 x y$$\stackrel{\text{assign}}{\to }$$$

Obtain the equations $S\=\left\{x\=x\left(u\,v\right)\,y\=y\left(u\,v\right)\right\}$ for the mapping $\left(u\,v\right)\to \left(x\,y\right)$ 

$u\=U\,v\=V$

$u=x^2-y^2,v=2xy$

$\stackrel{\text{solve (specified)}}{\to }$

$\left\{x=\frac{v}{2\mathrm{RootOf}\left(4{\mathrm{\_Z}}^4+4{\mathrm{\_Z}}^{2}u-v^2\right)}\,y=\mathrm{RootOf}\left(4{\mathrm{\_Z}}^4+4{\mathrm{\_Z}}^{2}u-v^2\right)\right\}$

$\stackrel{\text{all values}}{\to }$

$\left\{x=\frac{v}{\sqrt{-2u+2\sqrt{u^2+v^2}}}\,y=\frac{\sqrt{-2u+2\sqrt{u^2+v^2}}}{2}\right\},\left\{x=-\frac{v}{\sqrt{-2u+2\sqrt{u^2+v^2}}}\,y=-\frac{\sqrt{-2u+2\sqrt{u^2+v^2}}}{2}\right\},\left\{x=\frac{v}{\sqrt{-2u-2\sqrt{u^2+v^2}}}\,y=\frac{\sqrt{-2u-2\sqrt{u^2+v^2}}}{2}\right\},\left\{x=-\frac{v}{\sqrt{-2u-2\sqrt{u^2+v^2}}}\,y=-\frac{\sqrt{-2u-2\sqrt{u^2+v^2}}}{2}\right\}$

$\stackrel{\text{assign to a name}}{\to }$

$S$

Obtain the Jacobian

$\genfrac{}{}{0ex}{}{x}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S_1}$ = $\frac{v}{\sqrt{-2u+2\sqrt{u^2+v^2}}}$$\stackrel{\text{assign to a name}}{\to }$$X$$$$$

$\genfrac{}{}{0ex}{}{y}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S_1}$ = $\frac{\sqrt{-2u+2\sqrt{u^2+v^2}}}{2}$$\stackrel{\text{assign to a name}}{\to }$$Y$

$\left[U\,V\right]$

$\left[x^2-y^2\,2xy\right]$

$\stackrel{\text{Jacobian}}{\to }$

$4x^2+4y^2$

$\stackrel{\text{evaluate at point}}{\to }$

$\frac{4v^2}{-2u+2\sqrt{u^2+v^2}}-2u+2\sqrt{u^2+v^2}$

$\stackrel{\text{rationalize}}{\=}$

$-\frac{4\left(u\sqrt{u^2+v^2}-u^2-v^2\right)\left(\sqrt{u^2+v^2}+u\right)}{v^2}$

$\stackrel{\text{expand}}{\=}$

$4\sqrt{u^2+v^2}$

$\left[U\,V\right]$ = $\left[x^2-y^2\,2xy\right]$$\stackrel{\text{Jacobian}}{\to }$$4x^2\+4y^2$$\stackrel{\text{evaluate at point}}{\to }$$\frac{4v^2}{-2u\+2\sqrt{u^2\+v^2}}-2u\+2\sqrt{u^2\+v^2}$$\stackrel{\text{rationalize}}{\=}$$-\frac{4\left(u\sqrt{u^2\+v^2}-u^2-v^2\right)\left(u\+\sqrt{u^2\+v^2}\right)}{v^2}$$\stackrel{\text{expand}}{\=}$$4\sqrt{u^2\+v^2}$$$

Obtain the Jacobian matrix and the Jacobian from first principles

$\left[\begin{array}{cc}\frac{\partial }{\partial x} U& \frac{\partial }{\partial y} U\\ \frac{\partial }{\partial x} V& \frac{\partial }{\partial y} V\end{array}\right]$ = $\stackrel{\text{determinant}}{\to }$$4x^2\+4y^2$$\stackrel{\text{evaluate at point}}{\to }$$\frac{4v^2}{-2u\+2\sqrt{u^2\+v^2}}-2u\+2\sqrt{u^2\+v^2}$$\stackrel{\text{rationalize}}{\=}$$-\frac{4\left(u\sqrt{u^2\+v^2}-u^2-v^2\right)\left(\sqrt{u^2\+v^2}\+u\right)}{v^2}$$\stackrel{\text{expand}}{\=}$$4\sqrt{u^2\+v^2}$

Obtain the images of the edges of the region $R$ 

$\genfrac{}{}{0ex}{}{L_1}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S_1}$

$\frac{v^2}{-2u\+2\sqrt{u^2\+v^2}}\+\frac{1}{2}u-\frac{1}{2}\sqrt{u^2\+v^2}\=1$

$\stackrel{\text{move to left}}{\to }$

$\frac{v^2}{-2u\+2\sqrt{u^2\+v^2}}\+\frac{1}{2}u-\frac{1}{2}\sqrt{u^2\+v^2}-1\=0$

$\stackrel{\text{left hand side}}{\to }$

$\frac{v^2}{-2u\+2\sqrt{u^2\+v^2}}\+\frac{1}{2}u-\frac{1}{2}\sqrt{u^2\+v^2}-1$

$\stackrel{\text{rationalize}}{\=}$

$u-1$

$\stackrel{\text{equate to 0}}{\to }$

$u-1\=0$

$\genfrac{}{}{0ex}{}{L_2}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S_1}$

$\frac{v^2}{-2u\+2\sqrt{u^2\+v^2}}\+\frac{1}{2}u-\frac{1}{2}\sqrt{u^2\+v^2}\=4$

$\stackrel{\text{move to left}}{\to }$

$\frac{v^2}{-2u\+2\sqrt{u^2\+v^2}}\+\frac{1}{2}u-\frac{1}{2}\sqrt{u^2\+v^2}-4\=0$

$\stackrel{\text{left hand side}}{\to }$

$\frac{v^2}{-2u\+2\sqrt{u^2\+v^2}}\+\frac{1}{2}u-\frac{1}{2}\sqrt{u^2\+v^2}-4$

$\stackrel{\text{rationalize}}{\=}$

$u-4$

$\stackrel{\text{equate to 0}}{\to }$

$u-4\=0$

$\genfrac{}{}{0ex}{}{L_3}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S_1}$

$v\=1$

$\genfrac{}{}{0ex}{}{L_4}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{S_1}$

$v\=5$

Implement the integration over the region $R\prime$ 

${\int }_1^4{\int }_1^5\frac{f\left(x\left(u\,v\right)\,y\left(u\,v\right)\right)}{4\sqrt{u^2\+v^2}} \mathit{\ⅆ}v \mathit{\ⅆ}u$

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

$L≔\left[x^2-y^2\=1\, x^2-y^2\=4\,2 x y\=1\,2 x y\=5\right]\:$

$$\begin{gathered} U≔x^2-y^2\: \\ V≔2 x y\: \end{gathered}$$

Obtain the equations $S\=\left\{x\=x\left(u\,v\right)\,y\=y\left(u\,v\right)\right\}$ for the mapping $\left(u\,v\right)\to \left(x\,y\right)$ 

$S≔\mathrm{solve}\left(\left\{u\=U\,v\=V\right\}\,\left\{x\,y\right\}\,\mathrm{explicit}\right)$

$\left\{x\=\frac{v}{\sqrt{-2u\+2\sqrt{u^2\+v^2}}}\,y\=\frac{1}{2}\sqrt{-2u\+2\sqrt{u^2\+v^2}}\right\}\,\left\{x\=-\frac{v}{\sqrt{-2u\+2\sqrt{u^2\+v^2}}}\,y\=-\frac{1}{2}\sqrt{-2u\+2\sqrt{u^2\+v^2}}\right\}\,\left\{x\=\frac{v}{\sqrt{-2u-2\sqrt{u^2\+v^2}}}\,y\=\frac{1}{2}\sqrt{-2u-2\sqrt{u^2\+v^2}}\right\}\,\left\{x\=-\frac{v}{\sqrt{-2u-2\sqrt{u^2\+v^2}}}\,y\=-\frac{1}{2}\sqrt{-2u-2\sqrt{u^2\+v^2}}\right\}$

Obtain the Jacobian matrix and the Jacobian

$X≔\mathrm{eval}\left(x\,S_1\right)$

$\frac{v}{\sqrt{-2u\+2\sqrt{u^2\+v^2}}}$

$Y≔\mathrm{eval}\left(y\,S_1\right)$

$\frac{1}{2}\sqrt{-2u\+2\sqrt{u^2\+v^2}}$

Apply the Jacobian command in its two forms, one to get the matrix, and one to get the determinant.
Obtain $\frac{\partial \left(u\,v\right)}{\partial \left(x\,y\right)}$, the reciprocal of $\frac{\partial \left(x\,y\right)}{\partial \left(u\,v\right)}$.

Apply the rationalize and expand commands to $\frac{\partial \left(u\,v\right)}{\partial \left(x\,y\right)}$ to obtain an expression in $u$ and $v$.

$$\begin{gathered} \mathrm{Jacobian}\left(\left[U\,V\right]\,\left[x\,y\right]\right)\; \\ \mathrm{Jxy}≔\mathrm{Jacobian}\left(\left[U\,V\right]\,\left[x\,y\right]\,\mathrm{output}\=\mathrm{determinant}\right)\; \\ \mathrm{expand}\left(\mathrm{rationalize}\left(\mathrm{eval}\left(\mathrm{Jxy}\,\left[x\=X\,y\=Y\right]\right)\right)\right) \end{gathered}$$

Obtain the images of the boundaries of the region $R$ 

$$\begin{gathered} \mathrm{\λ}≔\mathrm{eval}\left(L_1\,S_1\right)\: \\ \mathrm{rationalize}\left(\mathrm{lhs}\left(\mathrm{\λ}\right)-\mathrm{rhs}\left(\mathrm{\λ}\right)\right)\=0 \end{gathered}$$

$u-1\=0$

$$\begin{gathered} \mathrm{\λ}≔\mathrm{eval}\left(L_2\,S_1\right)\: \\ \mathrm{rationalize}\left(\mathrm{lhs}\left(\mathrm{\λ}\right)-\mathrm{rhs}\left(\mathrm{\λ}\right)\right)\=0 \end{gathered}$$

$u-4\=0$

$$\begin{gathered} \mathrm{\λ}≔\mathrm{eval}\left(L_3\,S_1\right)\: \\ \mathrm{rationalize}\left(\mathrm{lhs}\left(\mathrm{\λ}\right)-\mathrm{rhs}\left(\mathrm{\λ}\right)\right)\=0 \end{gathered}$$

$v-1\=0$

$$\begin{gathered} \mathrm{\λ}≔\mathrm{eval}\left(L_4\,S_1\right)\: \\ \mathrm{rationalize}\left(\mathrm{lhs}\left(\mathrm{\λ}\right)-\mathrm{rhs}\left(\mathrm{\λ}\right)\right)\=0 \end{gathered}$$

$v-5\=0$

Implement the integration over the region $R\prime$ 

Use the Int command to generate the inert integral. Press the Enter key.

$\mathrm{Int}\left(\frac{f\left(x\left(u\,v\right)\,y\left(u\,v\right)\right)}{4\sqrt{u^2\+v^2}} \,\left[v\=1..5\,u\=1..4\right]\right)$

${\∫}_1^4{\∫}_1^5\frac{1}{4}\frac{f\left(x\left(u\,v\right)\,y\left(u\,v\right)\right)}{\sqrt{u^2\+v^2}}\ⅆv\ⅆu$