Question

Find theJacobian$\mathbf{\partial }\left(x,y\right)\mathbf{/}\mathbf{\partial }\left(u,y\right)$of the given transformations from variables x,yto variables u,v:

${}_{\mathbf{y}\mathbf{=}\mathbf{a}\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{h}\mathbf{}\mathbf{u}\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{v}\mathbf{,}}^{\mathbf{x}\mathbf{=}\mathbf{a}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{h}\mathbf{}\mathbf{u}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{v}\mathbf{,}}\mathbf{}\left(u\mathrm{and}v\mathrm{are}\mathrm{called}\mathrm{elliptic}\mathrm{cylinder}\mathrm{coordinates}\right)$

Short answer

The determinant of the Jacobian is $\left|J\right|=a^2\sin h^{2}u{\cos }^{2}v+a^2\cos h^{2}u{\sin }^{2}v$.

Step-by-step solution

Given Condition

The giventransformation is

$$\begin{gathered} x=a\cos hu\cos v \\ y=a\sin hu\sin v \end{gathered}$$.

Concept of the Jacobian

The Jacobian of x,ywith respect to s,t is

$$\begin{gathered} \mathit{J}\mathbf{=}\mathit{J}\left(\frac{x,y}{s,t}\right) \\ \mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{\partial }\left(x,y\right)}{\mathbf{\partial }\left(s,t\right)} \\ \mathbf{}\mathbf{}\mathbf{=}\left|\begin{array}{cc}\frac{\partial x}{\partial s}& \frac{\partial x}{\partial t}\\ \frac{\partial y}{\partial s}& \frac{\partial y}{\partial t}\end{array}\right| \end{gathered}$$

Determine the Jacobian

$$\begin{gathered} \left|J\right|=\frac{\partial (\mathrm{x},\mathrm{y})}{\partial (u,v)} \\ =\left|\begin{array}{cc}{\partial }_{u}x& {\partial }_{v}x\\ {\partial }_{u}y& {\partial }_{v}y\end{array}\right| \\ =\left|\begin{array}{cc}\frac{\partial \mathrm{x}}{\partial \mathrm{u}}& \frac{\partial \mathrm{x}}{\partial \mathrm{v}}\\ \frac{\partial \mathrm{y}}{\partial \mathrm{u}}& \frac{\partial \mathrm{y}}{\partial \mathrm{v}}\end{array}\right| \\ =\frac{\partial \mathrm{x}}{\partial \mathrm{u}}\frac{\partial \mathrm{y}}{\partial \mathrm{v}}-\frac{\partial \mathrm{y}}{\partial \mathrm{u}}\frac{\partial \mathrm{x}}{\partial \mathrm{v}} \end{gathered}$$

$=\left|\begin{array}{cc}a\sin hu\cos v& -a\cos hu\sin v\\ a\cos hu\sin v& a\sin hu\cos v\end{array}\right|$

$$\begin{gathered} =\left(a\sin hu\cos v\right)\left(a\sin hu\cos v\right)+\left(a\cos hu\sin v\right)\left(a\cos hu\sin v\right) \\ =a^2\sin h^{2}u{\cos }^2=a^2\cos h^{2}u{\sin }^{2}v \end{gathered}$$

The determinant of the Jacobian is $\left|\mathrm{J}\right|={\mathrm{a}}^2{\sinh }^2{\mathrm{ucos}}^2={\mathrm{a}}^2{\cosh }^2{\mathrm{usin}}^2\mathrm{v}$.