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'
Differentiating equation (i) partially with respective to 'v'
Differentiating equation (i) partially with respective to 'w'
Given y = 1 6 (u − v) …(ii)
Differentiating equation (ii) partially with respective to 'u'
Differentiating equation (ii) partially with respective to 'v'
Differentiating equation (ii) partially with respective to 'w'
Given z = 6uvw ..(iii)
Differentiating equation (iii) partially with respective to 'u'
Differentiating equation (iii) partially with respective to 'v'
Differentiating equation (iii) partially with respective to 'w'
<u>Step :-(ii)</u>
The Jacobian ∂(x, y, z)/ ∂(u, v, w) =
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
