Find the Jacobian ∂(x, y, z) ∂(u, v, w) for the indicated change of variables. If x = f(u, v, w), y = g(u, v, w), and z = h(u, v, w), then the Jacobian of x, y, and z with respect to u, v, and w is ∂(x, y, z) ∂(u, v, w) = ∂x ∂u ∂x ∂v ∂x ∂w ∂y ∂u ∂y ∂v ∂y ∂w ∂z ∂u ∂z ∂v ∂z ∂w . x = 1 6 (u + v), y = 1 6 (u − v), z = 6uvw

The Jacobian ∂(x, y, z) ∂(u, v, w) for the indicated change of variables

= -3072uv

Step-by-step explanation:

Step :-(i)

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'

Step :-(ii)

The Jacobian ∂(x, y, z)/ ∂(u, v, w) =

   Determinant       16(-16×6uv-0)-16(16×6uv)+0(0) = - 1536uv-1536uv

                                                                                 = -3072uv

Final answer:-

The Jacobian ∂(x, y, z)/ ∂(u, v, w) = -3072uv

im not quite sure if this is right but i got 5.6 oisutuve

step-by-step explanation:

d. bc = 12 and ef = 16

step-by-step explanation:

we are given that δabc : δdef such that .

also, it is given that

so, from the options, we need that .

a.

b.

c.  

d.

as, we can see that only option d gives the same ratio as i.e. .

hence, the possible lengths are bc = 12 and ef = 16.