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

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

$x\=\sqrt{\left(v\+u\right)\/2}$

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

$y\=\sqrt{\left(v-u\right)\/2}$

$v\=x^2\+y^2$

Table 5.6.3(a)   Mappings $R\leftrightarrow R\prime$

$\int {\int }_{R\prime }\left|\frac{\partial \left(x\,y\right)}{\partial \left(u\,v\right)}\right| \mathrm{dv} \mathrm{du}$

= ${\int }_1^4{\int }_9^{16}\frac{1}{4\sqrt{v^2-u^2}} \mathit{\ⅆ}v\mathit{\ⅆ}u$ 

$\=\frac{1}{4}\ln \left(\frac{9\+4\sqrt{5}}{16\+\sqrt{255}}\right)\+\ln \left(\frac{4\left(4\+\sqrt{15}\right)}{9\+\sqrt{65}}\right)\+\frac{9}{4}\left(\arcsin \left(\frac{1}{9}\right)-\arcsin \left(\frac{4}{9}\right)\right)\+4\left(\arcsin \left(\frac{1}{4}\right)-\arcsin \left(\frac{1}{16}\right)\right)$

$\doteq 0.443319590$

Initialize

Loading Student:-MultivariateCalculus

$L\=\left[x^2-y^2\=1\, x^2-y^2\=4\,x^2\+y^2\=9\,x^2\+y^2\=16\right]$$\stackrel{\text{assign}}{\to }$

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

$V\=x^2\+y^2$$\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 $\left(u\,v\right)\to \left(x\,y\right)$ 

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

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

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

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

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

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

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

$S$

Obtain the Jacobian matrix and the Jacobian

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

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

$\left[X\,Y\right]$ = $\left[\sqrt{\frac{v}{2}+\frac{u}{2}}\,\sqrt{-\frac{u}{2}+\frac{v}{2}}\right]$$\stackrel{\text{Jacobian}}{\to }$$\frac{1}{2\sqrt{2u+2v}\sqrt{-2u+2v}}$

Obtain the Jacobian matrix and the Jacobian from first principles

$\left[\begin{array}{cc}\frac{\partial }{\partial u} \mathrm{X}& \frac{\partial }{\partial v} \mathrm{X}\\ \frac{\partial }{\partial u} \mathrm{Y}& \frac{\partial }{\partial v} \mathrm{Y}\end{array}\right]$ = $\stackrel{\text{determinant}}{\to }$$\frac{1}{2\sqrt{2v\+2u}\sqrt{-2u\+2v}}$

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

Implement the integration over the region $R\prime$ 

${\int }_1^4{\int }_9^{16}\frac{1}{4\sqrt{v^2-u^2}} \mathit{\ⅆ}v \mathit{\ⅆ}u$

$\frac{1}{4}\ln \left(9\+4\sqrt{5}\right)\+\frac{9}{4}\arcsin \left(\frac{1}{9}\right)-\frac{1}{4}\ln \left(16\+\sqrt{5}\sqrt{3}\sqrt{17}\right)-4\arcsin \left(\frac{1}{16}\right)-\ln \left(9\+\sqrt{5}\sqrt{13}\right)-\frac{9}{4}\arcsin \left(\frac{4}{9}\right)\+2\ln \left(2\right)\+\ln \left(4\+\sqrt{5}\sqrt{3}\right)\+4\arcsin \left(\frac{1}{4}\right)$

$\stackrel{\text{at 10 digits}}{\to }$

$0.443319591$

Initialize

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

$L≔\left[x^2-y^2\=1\, x^2-y^2\=4\,x^2\+y^2\=9\,x^2\+y^2\=16\right]\:$

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

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

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

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

Obtain the Jacobian matrix and the Jacobian

$$\begin{gathered} T≔\mathrm{eval}\left(\left[x\,y\right]\,S_1\right)\; \\ \mathrm{Jacobian}\left(T\,\left[u\,v\right]\right)\; \\ J≔\mathrm{simplify}\left(\mathrm{combine}\left(\mathrm{Jacobian}\left(T\,\left[u\,v\right]\,\mathrm{output}\=\mathrm{determinant}\right)\right)\right) assuming \mathrm{positive} \end{gathered}$$

$\left[\frac{1}{2}\sqrt{2v\+2u}\,\frac{1}{2}\sqrt{-2u\+2v}\right]$

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

Implement the integration over the region $R\prime$ 

$$\begin{gathered} q≔\mathrm{Int}\left(J \,\left[v\=9..16\,u\=1..4\right]\right)\; \\ \mathrm{value}\left(q\right) \end{gathered}$$

${\∫}_1^4{\∫}_9^{16}\frac{1}{4\sqrt{-u^2\+v^2}}\ⅆv\ⅆu$