Question

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

$$\begin{gathered} \mathit{x}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\left(u^2-v^2\right) \\ \mathit{y}\mathbf{=}\mathit{u}\mathit{v} \end{gathered}$$

( u and v are called parabolic cylinder coordinates)

Short answer

The Jacobians is$\left|J\right|=u^2+v^2$

Step-by-step solution

Given Information

The given transformations from the variables x,y to variables u,v

$$\begin{gathered} x=\frac{1}{2}({\mathrm{u}}^2-{\mathrm{v}}^2) \\ y=uv \end{gathered}$$

( u and v are called parabolic cylinder coordinates)

 Step 2: Concept of Jacobian

The Jacobian determinant is used when making a change of variables when evaluating a multiple integral of a function over a region within its domain.

Evaluate Jacobian determinant

The determinant of the Jacobian is equal to:

$$\begin{gathered} \left|J\right|=\left|\begin{array}{cc}{\partial }_{u}x& {\partial }_{u}x\\ {\partial }_{u}y& {\partial }_{u}y\end{array}\right| \\ =\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}-\frac{\partial y}{\partial u}\frac{\partial x}{\partial v} \\ =u\times v-v\times \left(-v\right) \\ =u^2+v^2 \end{gathered}$$

Where ${\partial }_{a}b$is the partial derivative of b with respect to (or$\partial b/\partial a$ in other words).

Hence, the Jacobians is$\left|J\right|=u^2+v^2$