Chapter 15.9, Problem 1PT

Find the Jacobian for x = u²v², y = u² + v².

1. a) 2uv³ − 2u³v
2. b) 4u²v − 4uv²
3. c) 2u²v − 2uv²
4. d) 4uv³ − 4u³v

To determine

The appropriate option for the Jacobian for $x=u^2{v}^2,y=u^2+v^2$, from the given options $\left(\text{a}\right)\text{-}\left(\text{d}\right)$.

Answer to Problem 1PT

The appropriate option is $\underset{\_}{\left(\text{d}\right)}$.

Explanation of Solution

Formula used:

A transformation T is $C^1$ means that all the component functions have continuous first partial derivatives.

For a $C^1$ transformation T from $uv$-space to xy-space, the Jacobian of T is the determinant,

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

Calculation:

Given transformation is $x=u^2{v}^2,y=u^2+v^2$.

The Jacobian of the transformation $x=u^2{v}^2,y=u^2+v^2$ is calculated as follows.

$\begin{array}{c}\frac{\partial \left(x,y\right)}{\partial \left(u,v\right)}=\left|\begin{array}{cc}\frac{\partial }{\partial u}\left(u^2{v}^2\right)& \frac{\partial }{\partial v}\left(u^2{v}^2\right)\\ \frac{\partial }{\partial u}\left(u^2+v^2\right)& \frac{\partial }{\partial v}\left(u^2+v^2\right)\end{array}\right|\\ =\left|\begin{array}{cc}2u{v}^2& 2u^{2}v\\ 2u& 2v\end{array}\right|\\ =2u{v}^2\left(2v\right)-2u^{2}v\left(2u\right)\\ =4u{v}^3-4u^{3}v\end{array}$

Thus, $\frac{\partial \left(x,y\right)}{\partial \left(u,v\right)}=4u{v}^3-4u^{3}v$.

Therefore, the appropriate option is $\underset{\_}{\left(\text{d}\right)}$.