Answer:
The Jacobian ∂(x, y, z) ∂(u, v, w) for the indicated change of variables
= -3072uv
Step-by-step explanation:
<u>Step :-(i)</u>
Given x = 1 6 (u + v) …(i)
Differentiating equation (i) partially with respective to 'u'
![\frac{∂x}{∂u} = 16(1)+16(0)=16](https://tex.z-dn.net/?f=%5Cfrac%7B%E2%88%82x%7D%7B%E2%88%82u%7D%20%3D%2016%281%29%2B16%280%29%3D16)
Differentiating equation (i) partially with respective to 'v'
Differentiating equation (i) partially with respective to 'w'
![\frac{∂x}{∂w} = 0](https://tex.z-dn.net/?f=%5Cfrac%7B%E2%88%82x%7D%7B%E2%88%82w%7D%20%3D%200)
Given y = 1 6 (u − v) …(ii)
Differentiating equation (ii) partially with respective to 'u'
![\frac{∂y}{∂u} = 16(1) - 16(0)=16](https://tex.z-dn.net/?f=%5Cfrac%7B%E2%88%82y%7D%7B%E2%88%82u%7D%20%3D%2016%281%29%20-%2016%280%29%3D16)
Differentiating equation (ii) partially with respective to 'v'
![\frac{∂y}{∂v} = 16(0) - 16(1)= - 16](https://tex.z-dn.net/?f=%5Cfrac%7B%E2%88%82y%7D%7B%E2%88%82v%7D%20%3D%2016%280%29%20-%2016%281%29%3D%20-%2016)
Differentiating equation (ii) partially with respective to 'w'
![\frac{∂y}{∂w} = 0](https://tex.z-dn.net/?f=%5Cfrac%7B%E2%88%82y%7D%7B%E2%88%82w%7D%20%3D%200)
Given z = 6uvw ..(iii)
Differentiating equation (iii) partially with respective to 'u'
![\frac{∂z}{∂u} = 6vw](https://tex.z-dn.net/?f=%5Cfrac%7B%E2%88%82z%7D%7B%E2%88%82u%7D%20%3D%206vw)
Differentiating equation (iii) partially with respective to 'v'
![\frac{∂z}{∂v} =6 u (1)w=6uw](https://tex.z-dn.net/?f=%5Cfrac%7B%E2%88%82z%7D%7B%E2%88%82v%7D%20%3D6%20u%20%281%29w%3D6uw)
Differentiating equation (iii) partially with respective to 'w'
![\frac{∂z}{∂w} =6 uv(1)=6uv](https://tex.z-dn.net/?f=%5Cfrac%7B%E2%88%82z%7D%7B%E2%88%82w%7D%20%3D6%20uv%281%29%3D6uv)
<u>Step :-(ii)</u>
The Jacobian ∂(x, y, z)/ ∂(u, v, w) =
![\left|\begin{array}{ccc}16&16&0\\16&-16&0\\6vw&6uw&6uv\end{array}\right|](https://tex.z-dn.net/?f=%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D16%2616%260%5C%5C16%26-16%260%5C%5C6vw%266uw%266uv%5Cend%7Barray%7D%5Cright%7C)
Determinant 16(-16×6uv-0)-16(16×6uv)+0(0) = - 1536uv-1536uv
= -3072uv
<u>Final answer</u>:-
The Jacobian ∂(x, y, z)/ ∂(u, v, w) = -3072uv